If you assume that $X \to Y$ is finite, with $X$ a complex analytic space, then $Y$ a quasi-projective variety implies that $X$ is. This is a general version of Riemann's existence theorem, maybe due to Grauert and Remmert when $X$ and $Y$ are normal and to Grothendieck in general (perhaps with additional assumption that $X \to Y$ is unramified). (But I'm not sure if this is the correct attribution, and you should also probably look in the literature for the most general precise statements.)
If you just assume finite fibres, there is not much to say without making more restrictions. E.g. suppose that $X$ is an open subset of $Y$ (in the complex topology). Then the fibres of $X \to Y$ are finite (either singletons or empty), but $X$ won't be quasi-projective unless it is actually Zariski open in $Y$.
Since $X$ is smooth when $Y$ is, this shows that smoothness is not really the issue here.
Here are some speculations:
I don't know how the details would go, but I imagine that you can stratify $Y$ according to the nature of the fibres of the map $X \to Y$ (i.e. consider the strata along which the fibres of $X \to Y$ are of some fixed cardinality, counted with multiplicity). My guess would then be that $X$ is quasi-projective if and only this stratification is an algebraic stratification of $Y$. (The idea is that $X \to Y$ will be finite when restricted to the
strata, so Riemann existence would apply. The details could be tricky, though,
and my guess might be a little too naive.)