If $f: X \to Y$ is a finite map of compact complex manifolds, then we can push a cohomology class in $H^2(X)$ by considering the dual homology class in $H_{n-2}(X)$, pushing it forward, and then considering the dual of the push-forward in $H^2(Y)$. The question is: if class $a \in H^2(X)$ contains a kähler form, is it true that $f_* a$ also does? The answer is clearly positive when the map $f$ is unramified, in which case the kähler representative of $f_* a$ is just the "summing over the fibres" form, but what if the map is ramified?
passing to currents, it would be possible to construct a positive current on $Y$ as the push-forward of the kähler form on $X$, but I am not sure how to smooth it.
This is getting a bit long for a comment, so forgive me an answer.
To answer your original question (in a different way than you really wanted, I gather), the pushforward of a Kahler class $\omega$ by a finite map $f : X \to Y$ of compact manifolds is again a Kahler class. If $Z \subset Y$ is a closed analytic subspace of dimension $p$ then $$ \int_Z f_* \omega^p = \operatorname{deg} f_{|f^{-1}(Z)} \int_{f^{-1}(Z)} \omega^p > 0 $$ because $f^{-1}(Z)$ is again a closed analytic subspace of dimension $p$ and $\omega$ is Kahler. By Demailly-Paun's characterization of the Kahler cone, it follows that $f_*\omega$ is Kahler.
You asked for references on Kahler currents, and I gather that what you actually wanted to do was to prove the above directly (which should be doable, we're totally nuking a mosquito). The ones I know for Kahler currents in complex geometry are Demailly's book (Chapters 1 and 3) and notes (later chapters; it's advanced applications of everything in the book).
Specifically, Demailly discusses direct images of currents in Chapter 1.2 and positive currents in Chapter 3, and states without proof (but with pointers on how to assemble a proof) that direct images of a positive current by a proper map (which a finite map between compact manifolds surely is) is again a positive current of the same degree in Proposition 1.17 in Chapter 3. He also discusses smoothing of currents, but I don't have exact references to the parts that would be interesting to you.