[proofplan]
We identify the sheaf $\Omega_X^p$ of holomorphic $p$-forms as a finite-rank locally free $\mathcal{O}_X$-module, hence as a coherent analytic sheaf. Since $X$ is Stein, Cartan's Theorem B annihilates the positive sheaf cohomology groups $H^q(X,\Omega_X^p)$. Dolbeault's theorem identifies these sheaf cohomology groups with the Dolbeault cohomology groups $H^{p,q}_{\bar\partial}(X)$, and the quotient definition of Dolbeault cohomology then gives solvability of $\bar\partial u=\omega$ for every $\bar\partial$-closed form of positive antiholomorphic degree.
[/proofplan]
[step:Identify the holomorphic $p$-form sheaf as coherent]
Let $\mathcal{O}_X$ denote the sheaf of holomorphic functions on $X$, and let $\Omega_X^p$ denote the sheaf on $X$ assigning to each open set $U \subseteq X$ the $\mathcal{O}_X(U)$-module of holomorphic $p$-forms on $U$. Fix $p \in \{0,1,\ldots,n\}$.
For each point $a \in X$, choose a holomorphic coordinate chart
\begin{align*}
\varphi_a: U_a &\to V_a,
\end{align*}
where $U_a \subseteq X$ is open, $V_a \subseteq \mathbb{C}^n$ is open, and $\varphi_a$ is biholomorphic. For $j \in \{1,\ldots,n\}$, let $\pi_j:\mathbb{C}^n \to \mathbb{C}$ be the $j$-th coordinate projection, and define the coordinate function $z_{a,j}:U_a \to \mathbb{C}$ by $z_{a,j}=\pi_j\circ \varphi_a$.
Let $\mathcal{I}_p$ denote the set of strictly increasing $p$-tuples
\begin{align*}
\mathcal{I}_p &= \{(i_1,\ldots,i_p):1 \leq i_1 < \cdots < i_p \leq n\},
\end{align*}
with the convention that $\mathcal{I}_0=\{\emptyset\}$. For $I=(i_1,\ldots,i_p)\in \mathcal{I}_p$, define
\begin{align*}
dz_{a,I} &= dz_{a,i_1}\wedge \cdots \wedge dz_{a,i_p} \in \Omega_X^p(U_a),
\end{align*}
and for $p=0$ define $dz_{a,\emptyset}=1 \in \mathcal{O}_X(U_a)=\Omega_X^0(U_a)$.
By the coordinate description of holomorphic differential forms, for every open set $W \subseteq U_a$ and every section $\alpha \in \Omega_X^p(W)$, there are unique holomorphic functions $f_I \in \mathcal{O}_X(W)$, indexed by $I \in \mathcal{I}_p$, such that
\begin{align*}
\alpha &= \sum_{I\in \mathcal{I}_p} f_I\, dz_{a,I}|_W.
\end{align*}
Thus $\Omega_X^p|_{U_a}$ is a free $\mathcal{O}_X|_{U_a}$-module with basis $\{dz_{a,I}:I\in \mathcal{I}_p\}$. Since $\#\mathcal{I}_p=\binom{n}{p}$, the sheaf $\Omega_X^p$ is locally free of rank $\binom{n}{p}$.
Applying Finite-Rank Locally Free Analytic Sheaves Are Coherent to the finite-rank locally free $\mathcal{O}_X$-module $\Omega_X^p$, we conclude that $\Omega_X^p$ is coherent.
[guided]
We first isolate the sheaf to which Cartan's Theorem B will be applied. Let $\mathcal{O}_X$ be the sheaf of holomorphic functions on $X$, and let $\Omega_X^p$ be the sheaf whose value on an open set $U\subseteq X$ is the $\mathcal{O}_X(U)$-module of holomorphic $p$-forms on $U$. The goal of this step is to verify the coherence hypothesis needed for Cartan's Theorem B.
Fix $a\in X$. Since $X$ is a complex manifold of complex dimension $n$, there is a holomorphic coordinate chart
\begin{align*}
\varphi_a: U_a &\to V_a,
\end{align*}
where $U_a\subseteq X$ is open, $V_a\subseteq \mathbb{C}^n$ is open, and $\varphi_a$ is biholomorphic. For $j\in\{1,\ldots,n\}$, let $\pi_j:\mathbb{C}^n\to\mathbb{C}$ be the $j$-th coordinate projection, and define $z_{a,j}:U_a\to\mathbb{C}$ by $z_{a,j}=\pi_j\circ\varphi_a$.
The local basis for $p$-forms is indexed by increasing choices of $p$ coordinate directions. Define
\begin{align*}
\mathcal{I}_p &= \{(i_1,\ldots,i_p):1 \leq i_1 < \cdots < i_p \leq n\},
\end{align*}
and use the convention $\mathcal{I}_0=\{\emptyset\}$. For $I=(i_1,\ldots,i_p)\in\mathcal{I}_p$, set
\begin{align*}
dz_{a,I} &= dz_{a,i_1}\wedge\cdots\wedge dz_{a,i_p}\in\Omega_X^p(U_a),
\end{align*}
while for $p=0$ set $dz_{a,\emptyset}=1\in\mathcal{O}_X(U_a)=\Omega_X^0(U_a)$.
Now take any open set $W\subseteq U_a$ and any holomorphic $p$-form $\alpha\in\Omega_X^p(W)$. In holomorphic coordinates, every holomorphic $p$-form has a unique expansion
\begin{align*}
\alpha &= \sum_{I\in\mathcal{I}_p} f_I\, dz_{a,I}|_W,
\end{align*}
where each coefficient $f_I$ lies in $\mathcal{O}_X(W)$. This proves that $\Omega_X^p|_{U_a}$ is a free $\mathcal{O}_X|_{U_a}$-module with basis $\{dz_{a,I}:I\in\mathcal{I}_p\}$.
There are exactly $\binom{n}{p}$ increasing $p$-tuples in $\mathcal{I}_p$, so $\Omega_X^p$ is locally free of finite rank $\binom{n}{p}$. The theorem Finite-Rank Locally Free Analytic Sheaves Are Coherent applies because $\Omega_X^p$ is a finite-rank locally free $\mathcal{O}_X$-module on the complex manifold $X$. Therefore $\Omega_X^p$ is a coherent analytic sheaf.
[/guided]
[/step]
[step:Apply Cartan's Theorem B to the coherent sheaf $\Omega_X^p$]
Let $q\geq 1$. The theorem Cartan's Theorem B states that if $Y$ is a Stein manifold and $\mathcal{F}$ is a coherent analytic sheaf on $Y$, then
\begin{align*}
H^r(Y,\mathcal{F}) &= 0
\end{align*}
for every $r\geq 1$. We apply it with $Y=X$, $\mathcal{F}=\Omega_X^p$, and $r=q$. The hypothesis that $Y$ is Stein holds by assumption, and the coherence of $\Omega_X^p$ was proved in the previous step. Hence
\begin{align*}
H^q(X,\Omega_X^p) &= 0.
\end{align*}
[guided]
We now use the defining cohomological strength of Stein manifolds. The theorem Cartan's Theorem B says that for a Stein manifold $Y$ and a coherent analytic sheaf $\mathcal{F}$ on $Y$, every positive sheaf cohomology group vanishes:
\begin{align*}
H^r(Y,\mathcal{F}) &= 0
\end{align*}
for all $r\geq 1$.
We verify its hypotheses in the present situation. First, $X$ is Stein by the theorem statement. Second, the sheaf $\Omega_X^p$ is coherent by the preceding step. Third, the integer $q$ satisfies $q\geq 1$ by hypothesis. Therefore Cartan's Theorem B applies with $Y=X$, $\mathcal{F}=\Omega_X^p$, and $r=q$, giving
\begin{align*}
H^q(X,\Omega_X^p) &= 0.
\end{align*}
This is the sheaf-cohomological vanishing that will be transported to Dolbeault cohomology in the next step.
[/guided]
[/step]
[step:Use Dolbeault's theorem to transfer the sheaf cohomology vanishing]
Let $\mathcal{E}_X^{p,r}$ denote the sheaf on $X$ assigning to each open set $U\subseteq X$ the complex vector space of smooth $(p,r)$-forms on $U$. For every integer $r\geq 0$, let
\begin{align*}
\bar\partial_{p,r}: \mathcal{E}_X^{p,r} &\to \mathcal{E}_X^{p,r+1}
\end{align*}
denote the sheaf morphism induced by the Dolbeault operator, and let
\begin{align*}
\iota_p:\Omega_X^p &\to \mathcal{E}_X^{p,0}
\end{align*}
denote the inclusion of holomorphic $p$-forms as smooth $(p,0)$-forms.
By Dolbeault's Theorem, because $X$ is a complex manifold and $p\in\{0,\ldots,n\}$, the sequence
\begin{align*}
0 \to \Omega_X^p \xrightarrow{\iota_p} \mathcal{E}_X^{p,0}
\xrightarrow{\bar\partial_{p,0}} \mathcal{E}_X^{p,1}
\xrightarrow{\bar\partial_{p,1}} \mathcal{E}_X^{p,2}
\xrightarrow{\bar\partial_{p,2}} \cdots
\end{align*}
is a fine resolution of $\Omega_X^p$. Therefore the induced Dolbeault comparison map
\begin{align*}
D_{p,q}: H^{p,q}_{\bar\partial}(X) &\to H^q(X,\Omega_X^p)
\end{align*}
is an isomorphism of complex vector spaces. Combining this isomorphism with
\begin{align*}
H^q(X,\Omega_X^p) &= 0
\end{align*}
from Cartan's Theorem B gives
\begin{align*}
H^{p,q}_{\bar\partial}(X) &= 0.
\end{align*}
Since $p\in\{0,1,\ldots,n\}$ and $q\geq 1$ were arbitrary, this proves the stated vanishing for all such $p$ and $q$.
[guided]
Cartan's Theorem B gives vanishing for sheaf cohomology. The theorem we want is a statement about Dolbeault cohomology, so we need the bridge between these two cohomology theories.
Define $\mathcal{E}_X^{p,r}$ to be the sheaf of smooth $(p,r)$-forms on $X$. For each $r\geq 0$, define the sheaf morphism
\begin{align*}
\bar\partial_{p,r}: \mathcal{E}_X^{p,r} &\to \mathcal{E}_X^{p,r+1}
\end{align*}
to be the Dolbeault operator on smooth forms. Also define
\begin{align*}
\iota_p:\Omega_X^p &\to \mathcal{E}_X^{p,0}
\end{align*}
to be the inclusion sending a holomorphic $p$-form to the same differential form regarded as a smooth $(p,0)$-form.
The theorem Dolbeault's Theorem applies to every complex manifold and every holomorphic degree $p$ in the range $0\leq p\leq n$. These hypotheses hold here because $X$ is a Stein manifold, hence a complex manifold, and $p\in\{0,\ldots,n\}$. Dolbeault's theorem states that the complex
\begin{align*}
0 \to \Omega_X^p \xrightarrow{\iota_p} \mathcal{E}_X^{p,0}
\xrightarrow{\bar\partial_{p,0}} \mathcal{E}_X^{p,1}
\xrightarrow{\bar\partial_{p,1}} \mathcal{E}_X^{p,2}
\xrightarrow{\bar\partial_{p,2}} \cdots
\end{align*}
is a fine resolution of $\Omega_X^p$, and consequently it computes the sheaf cohomology of $\Omega_X^p$.
In concrete terms, this gives an isomorphism of complex vector spaces
\begin{align*}
D_{p,q}: H^{p,q}_{\bar\partial}(X) &\to H^q(X,\Omega_X^p).
\end{align*}
From the previous step, the target is the zero vector space:
\begin{align*}
H^q(X,\Omega_X^p) &= 0.
\end{align*}
Since $D_{p,q}$ is an isomorphism, its domain must also be the zero vector space. Therefore
\begin{align*}
H^{p,q}_{\bar\partial}(X) &= 0.
\end{align*}
Because the argument used only the assumptions $p\in\{0,1,\ldots,n\}$ and $q\geq 1$, the vanishing holds for every such pair $(p,q)$.
[/guided]
[/step]
[step:Translate the zero Dolbeault cohomology group into solvability of $\bar\partial u=\omega$]
Fix $p\in\{0,1,\ldots,n\}$ and $q\geq 1$. Define the closed and exact smooth form spaces
\begin{align*}
Z^{p,q}_{\bar\partial}(X)
&= \{\eta\in \mathcal{E}^{p,q}(X):\bar\partial_{p,q}\eta=0\},\\
B^{p,q}_{\bar\partial}(X)
&= \{\bar\partial_{p,q-1}v:v\in \mathcal{E}^{p,q-1}(X)\}.
\end{align*}
By definition,
\begin{align*}
H^{p,q}_{\bar\partial}(X) &= Z^{p,q}_{\bar\partial}(X)/B^{p,q}_{\bar\partial}(X).
\end{align*}
The previous step gives $H^{p,q}_{\bar\partial}(X)=0$, so this quotient is the zero vector space. Hence
\begin{align*}
Z^{p,q}_{\bar\partial}(X) &= B^{p,q}_{\bar\partial}(X).
\end{align*}
Let $\omega\in \mathcal{E}^{p,q}(X)$ be $\bar\partial$-closed. This means $\bar\partial_{p,q}\omega=0$, so $\omega\in Z^{p,q}_{\bar\partial}(X)$. Since $Z^{p,q}_{\bar\partial}(X)=B^{p,q}_{\bar\partial}(X)$, there exists $u\in \mathcal{E}^{p,q-1}(X)$ such that
\begin{align*}
\bar\partial_{p,q-1}u &= \omega.
\end{align*}
This is exactly the asserted solvability of $\bar\partial u=\omega$.
[guided]
The vanishing statement says that a certain quotient vector space is zero. We now unpack that quotient to obtain the explicit solution of the $\bar\partial$-equation.
Fix $p\in\{0,1,\ldots,n\}$ and $q\geq 1$. The condition $q\geq 1$ ensures that the degree $q-1$ appearing in the unknown form is nonnegative. Define
\begin{align*}
Z^{p,q}_{\bar\partial}(X)
&= \{\eta\in \mathcal{E}^{p,q}(X):\bar\partial_{p,q}\eta=0\},
\\
B^{p,q}_{\bar\partial}(X)
&= \{\bar\partial_{p,q-1}v:v\in \mathcal{E}^{p,q-1}(X)\}.
\end{align*}
The space $Z^{p,q}_{\bar\partial}(X)$ consists of $\bar\partial$-closed smooth $(p,q)$-forms, while $B^{p,q}_{\bar\partial}(X)$ consists of $\bar\partial$-exact smooth $(p,q)$-forms.
By the definition of Dolbeault cohomology,
\begin{align*}
H^{p,q}_{\bar\partial}(X) &= Z^{p,q}_{\bar\partial}(X)/B^{p,q}_{\bar\partial}(X).
\end{align*}
The vanishing already proved gives
\begin{align*}
H^{p,q}_{\bar\partial}(X) &= 0.
\end{align*}
Thus every class in the quotient is zero, which is equivalent to saying that every $\bar\partial$-closed smooth $(p,q)$-form is already $\bar\partial$-exact:
\begin{align*}
Z^{p,q}_{\bar\partial}(X) &= B^{p,q}_{\bar\partial}(X).
\end{align*}
Now let $\omega\in\mathcal{E}^{p,q}(X)$ be $\bar\partial$-closed. By definition of $\bar\partial$-closedness,
\begin{align*}
\bar\partial_{p,q}\omega &= 0,
\end{align*}
so $\omega\in Z^{p,q}_{\bar\partial}(X)$. Since $Z^{p,q}_{\bar\partial}(X)=B^{p,q}_{\bar\partial}(X)$, the form $\omega$ belongs to $B^{p,q}_{\bar\partial}(X)$. By the definition of $B^{p,q}_{\bar\partial}(X)$, there exists a smooth form $u\in\mathcal{E}^{p,q-1}(X)$ such that
\begin{align*}
\bar\partial_{p,q-1}u &= \omega.
\end{align*}
This is precisely the assertion that $\bar\partial u=\omega$.
[/guided]
[/step]
