  We show persistence of both Anderson and dynamical localization in
Schr\"odinger operators with non-positive (attractive) random decaying
potential. We consider an Anderson-type Schr\"odinger operator with a
non-positive ergodic random potential, and multiply the random potential by a
decaying envelope function. If the envelope function decays slower than
$|x|^{-2}$ at infinity, we prove that the operator has infinitely many
eigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity,
we determine the number of bound states below a given energy $E<0$,
asymptotically as $\alpha\downarrow 0$. To show that bound states located at
the bottom of the spectrum are related to the phenomenon of Anderson
localization in the corresponding ergodic model, we prove: (a) these states are
exponentially localized with a localization length that is uniform in the decay
exponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$.
