As Taras Banakh noted, there is a theorem for Tychonoff powers,
Theorem Let $\kappa\geq \omega$. If $X^{\omega}$ is Baire, then $X^{\kappa}$ is Baire, where the powers are considered in the Tychonoff product.
Now, if we assume Scheepers' theorem we have the following
Corollary Let $X$ be a topological space, if for all cardinal $\kappa$, the box power $X^{\kappa}$ is a Baire space then the Tychonoff power $X^{\omega}$ is a Baire space.