  Let $f$ be a transcendental entire function and $U$ be a Fatou component of
$f$. We show that if $U$ is an escaping wandering domain of $f$, then most
boundary points of $U$ (in the sense of harmonic measure) are also escaping. In
the other direction we show that if enough boundary points of $U$ are escaping,
then $U$ is an escaping Fatou component. Some applications of these results are
given; for example, if $I(f)$ is the escaping set of $f$, then
$I(f)\cup\{\infty\}$ is connected.
