General theorems concerning the existence and uniqueness of invariant measures are proved for a certain class of regular diffusion processes in Separable Banach spaces under some weak compactness and other conditions. Then, based on these theorems, some verifiable sufficient conditions are obtained to ensure the existence and uniqueness of an invariant distribution for the strong solution to some nonlinear evolution equations in a Hilbert space. The results are applied to certain monotone parabolic Itô equations as well as to the 2-D Navier-Stokes equations under random perturbations