As in Mark Meckes's comment, this is equivalent to log-Sobolev for standard Gaussian measure on $\mathbb{R}^\mathbb{N}$, which in turn follows immediately from the finite-dimensional case. In order to show the log-Sobolev inequality
$$\int |f|^2 \ln |f| \le \int \|\nabla f\| + \frac{1}{2} \int |f|^2 \ln \int |f|^2$$
it is sufficient to prove it for smooth cylinder functions $f \in L^2(\mathbb{R}^\mathbb{N}, \mu)$, i.e. which depend on only finitely many coordinates. But for $f$ depending on $n$ coordinates, this is precisely the log-Sobolev inequality for standard Gaussian measure on $\mathbb{R}^n$. (Unlike the classical Sobolev inequality, there is no dimension-dependent constant!)

Leonard Gross's original paper introducing the log-Sobolev inequality already mentions the extension to infinite dimensions (though it does not work it out explicitly).

Gross, Leonard. Logarithmic Sobolev Inequalities. *American Journal of Mathematics* **97** No. 4 (Winter, 1975), pp. 1061-1083.