Let $\{\langle X_\alpha,\mathscr{T}_\alpha\rangle: \alpha \in A\}$ be a family of topological spaces, let $X = \prod_{\alpha\in A} X_\alpha,$ and let $\mathscr{T}$ be the product topology on $X$. Let $\mathscr{F} = \{X\setminus V:V \in \mathscr{T}\}$, the family of closed sets in $\langle X,\mathscr{T}\rangle$. For each $\alpha \in A$ let $\pi_\alpha:X\to X_\alpha$ be the projection map, and let $\mathscr{F}_\alpha$ be the set of closed subsets of $X_\alpha$.
For $\alpha \in A$ let $\mathscr{S}_\alpha = \{\pi_\alpha^{-1}[U]:U \in \mathscr{T}_\alpha\}$, let $\mathscr{C}_\alpha = \{\pi_\alpha^{-1}[F]:F \in \mathscr{F}_\alpha\} = \{X\setminus V:V\in\mathscr{S}_\alpha\}$, and let $\mathscr{S} = \bigcup_{\alpha\in A}\mathscr{S}_\alpha\text{ and }\mathscr{C} = \bigcup_{\alpha\in A}\mathscr{C}_\alpha.$ $\mathscr{S}$ is a subbase for $\mathscr{T}$, so $\mathscr{C}$ is a closed subbase for $\mathscr{F}$. That is, every $V\in \mathscr{T}$ is a union of sets that are intersections of finitely many members of $\mathscr{S}$, so every $F\in \mathscr{F}$ is (by De Morgan’s laws) an intersection of sets that are unions of finitely many members of $\mathscr{C}$.
A union of finitely many members of $\mathscr{C}$ has the form $\bigcup_{\alpha\in \Phi} \pi_\alpha^{-1}[F_\alpha],$ where $\Phi$ is a finite subset of $A$, and $F_\alpha\in\mathscr{F}_\alpha$ for each $\alpha \in \Phi$. Thus, an arbitrary closed set in $X$ can be written in the form $\bigcap_{\xi\in\Xi}\ \bigcup_{\alpha\in\Phi_\xi}\pi_\alpha^{-1}F_{\xi,\alpha},$ where $\Xi$ is some index set, $\Phi_xi$ is a finite subset of $A$ for each $\xi\in\Xi$, and $F_{\xi,\alpha}\in\mathscr{F}_\alpha$ for each $\xi\in\Xi$ and $\alpha\in\Phi_\xi$.
Unless you limit yourself to products of finitely many spaces, you’ll have a hard time finding a simpler general form for the closed sets.