I've unaccepted kjetil's answer since, as was pointed out in the comments, it assumes $X$ and $Y$ are independent.

The following answer should work when $X$ and $Y$ are dependent, by using whuber's suggestion:

\begin{align}
\text{Var}(XY) &= E((XY)^2) - E(XY)^2 \\
&\le E(X^2Y^2) \\
&\le E(X^2)\sup(Y^2) \\
&= E(X^2) \\
&= \text{Var}(X) + E(X)^2 \\
&< \infty 
\end{align}

Note that the result also holds for any bounded $Y$ (since $\sup(Y^2)$ will be finite).