We will prove this by induction on $d$. The case $d = 0$ holds because in that case $X$ is the spectrum of a finite dimensional $k$-algebra $A$ (Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0}) and every coherent sheaf $\mathcal{F}$ corresponds to a finite $A$-module $M = H^0(X, \mathcal{F})$ which has $\dim_k M < \infty$. \medskip\noindent Assume $d > 0$ and the result has been shown for separated schemes of finite type of dimension $< d$. The scheme $X$ is Noetherian. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow \dim_k H^d(X, \mathcal{F}) < \infty $$ We are going to use the result of Cohomology of Schemes, Lemma \ref{coherent-lemma-property-initial} to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$. \medskip\noindent Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of cohomology $$ H^d(X, \mathcal{F}_1) \to H^d(X, \mathcal{F}) \to H^d(X, \mathcal{F}_2) $$ Thus if $\mathcal{P}$ holds for $\mathcal{F}_1$ and $\mathcal{F}_2$, then it holds for $\mathcal{F}$. \medskip\noindent Let $Z \subset X$ be an integral closed subscheme. Let $\mathcal{I}$ be a coherent sheaf of ideals on $Z$. To finish the proof we have to show that $H^d(X, i_*\mathcal{I}) = H^d(Z, \mathcal{I})$ is finite dimensional. If $\dim(Z) < d$, then the result holds because the cohomology group will be zero (Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}). In this way we reduce to the situation discussed in the following paragraph. \medskip\noindent Assume $X$ is a variety of dimension $d$ and  $\mathcal{F} = \mathcal{I}$ is a coherent ideal sheaf. In this case we have a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0 $$ where $i : Z \to X$ is the closed subscheme defined by $\mathcal{I}$. By induction hypothesis we see that $H^{d - 1}(Z, \mathcal{O}_Z) = H^{d - 1}(X, i_*\mathcal{O}_Z)$ is finite dimensional. Thus we see that it suffices to prove the result for the structure sheaf. \medskip\noindent We can apply Chow's lemma (Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian}) to the morphism $X \to \Spec(k)$. Thus we get a diagram $$ \xymatrix{ X \ar[rd]_g & X' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_k \ar[dl] \\ & \Spec(k) & } $$ as in the statement of Chow's lemma. Also, let $U \subset X$ be the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism. We may assume $X'$ is a variety as well, see Cohomology of Schemes, Remark \ref{coherent-remark-chow-Noetherian}. The morphism $i' = (i, \pi) : X' \to \mathbf{P}^n_X$ is a closed immersion (loc. cit.). Hence $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_k}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^n_X}(1) $$ is $\pi$-relatively ample (for example by Morphisms, Lemma \ref{morphisms-lemma-characterize-ample-on-finite-type}). Hence by Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting} there exists an $n \geq 0$ such that $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$. Choose any nonzero global section $s$ of $\mathcal{L}^{\otimes n}$. Since $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$, the section $s$ corresponds to section of $\mathcal{G}$, i.e., a map $\mathcal{O}_X \to \mathcal{G}$. Since $s|_U \not = 0$ as $X'$ is a variety and $\mathcal{L}$ invertible, we see that $\mathcal{O}_X|_U \to \mathcal{G}|_U$ is nonzero. As $\mathcal{G}|_U = \mathcal{L}^{\otimes n}|_{\pi^{-1}(U)}$ is invertible we conclude that we have a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where $\mathcal{Q}$ is coherent and supported on a proper closed subscheme of $X$. Arguing as before using our induction hypothesis, we see that it suffices to prove $\dim H^d(X, \mathcal{G}) < \infty$. \medskip\noindent By the Leray spectral sequence (Cohomology, Lemma \ref{cohomology-lemma-apply-Leray}) we see that $H^d(X, \mathcal{G}) = H^d(X', \mathcal{L}^{\otimes n})$. Let $\overline{X}' \subset \mathbf{P}^n_k$ be the closure of $X'$. Then $\overline{X}'$ is a projective variety of dimension $d$ over $k$ and $X' \subset \overline{X}'$ is a dense open. The invertible sheaf $\mathcal{L}$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X$. By Cohomology, Proposition \ref{cohomology-proposition-cohomological-dimension-spectral} the map $$ H^d(\overline{X}', \mathcal{O}_{\overline{X}'}(n)) \longrightarrow H^d(X', \mathcal{L}^{\otimes n}) $$ is surjective. Since the cohomology group on the left has finite dimension by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-projective} the proof is complete.