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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 (add a comment if you want a proof sketch of this), we conclude that the state space of a von Neumann algebra $A$ is (weak-*) first countable iff $A$ is finite-dimensional.