Here is my own attempt. We write $(F,G)\colon C\leftrightarrows D$ for the given Quillen equivalence and $(F^X,G^X)\colon C^X\leftrightarrows D^X$ for the pointwise induced adjunction. First we check that $(F^X,G^X)$ is a Quillen adjunction. It suffices to show that $G^X$ preserves fibrations and trivial fibrations which is clear because of the pointwise definition of fibrations and weak equivalences. It remains to show that $(F^X,G^X)$ is a Quillen equivalence, that is, a map $F^X(c^X)\to d^X$ in $D^X$ with $c^X$ cofibrant and $d^X$ fibrant is a weak equivalence if and only if the adjoint map $c^X\to G^X(d^X)$ is a weak equivalence in $C^X$. This follows from the pointwise definition of weak equivalences and fibrations once we know that cofibrant objects in $C^X$ are pointwise cofibrant.
That us check the last claim. Assume $c^X$ is cofibrant, that is, the induced map $L_x c^X\to c^X_x$ is a cofibration for every $x\in X$, where $L_x$ denotes the latching space functor. To see that $c^X$ is pointwise cofibrant it is enough to check that $L_x c^X$ is cofibrant for every $x\in X$. I don't see that yet.