Every von Neumann algebra is a C$^*$-algebra. So the usual theorem that a C$^*$-algebra $A$ is (norm) separable iff its state space is first countable in the weak-* topology (i.e. the topology $\sigma(A^*,A)$) applies. As von Neumann algebras are norm separable iff they are finite-dimensional, we conclude that the state space of a von Neumann algebra $A$ is (weak-*) first countable iff $A$ is finite-dimensional.
Added in edit
On reflection, the proof I thought of for first countability didn't work. However, I don't know of any compact Hausdorff first-countable space that isn't second countable, so I'll leave that open for others.
I will instead show it for second countability. Unaccept this answer if you think it's no longer adequate with that restriction. As all von Neumann algebras are unital, we can restrict to the case of a unital C$^*$-algebra $A$. Suppose that the state space $X$ is second countable. As $A$ is unital, $X$ is also compact, so by Urysohn's metrization theorem, $X$ is metrizable. Therefore the C$^*$-algebra $C(X)$ is separable (see Conway's Functional Analysis, chapter V, Theorem 6.6). By a theorem of Kadison, the map $\zeta : A \rightarrow C(X)$ defined by $$ \zeta(a)(\phi) = \phi(x) $$ embeds $A$ as a closed subspace (but not subalgebra) of $C(X)$. Specifically, the image of $\zeta$ is the continuous affine functions (see Lemma 2.5 of Kadison's A Representation Theory for Commutative Topological Algebra. This only does it for the self-adjoint elements and real-valued functions, but each element $a$ of $A$ can be uniquely expressed as $a = a_r + ia_i$ where $a_r,a_i$ are self-adjoint, and this is enough to extend it to all elements of $A$ and complex-valued functions). As a subspace of a separable metrizable space is separable, this shows that $A$ is separable.
For the statement that von Neumann algebras are finite-dimensional iff they are norm separable, I have found that this has been asked and answered before here. It is proved in Corollary 1.3.17 of this thesis.