In this paper, we prove that in many cases the random attractor defined and in [2] and [4] has finite Hausdorff dimension. We generalize a method used in [8] and [1] in the determistic case, the generalization is made through an ergodicity argument. The main assumption amounts to say that the diameter of the attractor has finite expectation. We apply our results to a stochastic reaction-diffusion equation and to a stochastic nonlinear wave equation. Other examples could be considered