In the thin domain Oε=T2×(0,ε), where T2 is a two-dimensional torus, we consider the 3D Navier-Stokes equations, perturbed by a white in time random force, and the Leray αα-approximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit ε→0. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure μμ comprises asymptotical in time statistical properties of solutions for the 3D Navier-Stokes equations in Oε, when ε≪1.