In the pointed Borel construction, you clearly mean $\wedge$
and not $\times$. Thus $$EG_+\wedge_G X = EG\times_G X/EG\times_G\ast.$$
Out of laziness, I'll assume that your $X$ and $Y$ are of the $G$-homotopy
types of $G$-CW complexes. The
based $G$-map $id\times f\colon EG\times X\longrightarrow EG\times Y$
is a homotopy equivalence on passage to $H$-fixed points for all
$H\subset G$: the condition is empty unless $H=e$, when it is your
hypothesis. Therefore $id\times f$ is a $G$-homotopy equivalence.
Via the inclusions of $EG$ in source and target given by the
basepoints of $X$ and $Y$, $id\times f$ is a map over $EG$ and
therefore a $G$-homotopy equivalence over $EG$ since the inclusions
of $EG$ in source and target are $G$-cofibrations by your well-pointed hypothesis. On passage to
orbits over $G$ and quotient spaces, it follows that
$$id\wedge_G f\colon EG_+\wedge_G X \longrightarrow EG_+\wedge_G Y$$
is a based homotopy equivalence.