We describe the support of the distribution for a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Both inclusions are proved by means of a general Wong--Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.