The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)
ADDED: I'm far from being an expert on the subject, but maybe I can focus the discussion a little more. The notion of finite $W$-algebra is tricky to define (there being at least three equivalent but different-looking definitions in the literature). Essentially it's an associative algebra attached to a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e \in \mathfrak{g}$. Usually the case $e=0$ is ignored, but the convention then is that the resulting finite $W$-algebra is just the universal enveloping algebra $U(\mathfrak{g})$. Kostant originally studied the opposite extreme when $e$ is regular ("principal") and the resulting algebra is isomorphic to the center of $U(\mathfrak{g})$. Premet extended the definition, using an $\mathfrak{sl}_2$-triple for (nonzero) $e$.
In any case, the finite dimensional representations are the main object of study; even their existence is problematic at first. Only in rank 1 are the results given by classical methods, while in general there are close connections with the primitive ideals of the universal enveloping algebra. Besides the work I mentioned above I should have pointed out the papers by Premet and by Ivan Losev: see for example Losev's recent ICM talk here and his earlier paper here.
