In the framework of stochastic non-intrusive finite element modeling, a common practice is using Monte Carlo simulation. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. The different Monte Carlo samplings correspond to realizations of the random variables characterizing the stochastic behavior of the model. Thus, this requires solving a set deterministic problems with the same structure, that is with variations concerning the material parameters and the loading data. Consequently, the different problems to be solved are in practice similar to each other. The reduced basis strategy is therefore a sensible option to reduce computational cost, provided that the quality of the numerical solution is guaranteed. The paper introduces a goal-oriented strategy allowing to successively enrich the reduced basis along the Monte Carlo process. The method is based on assessing the error of the reduced basis solution with a residual estimate for the prescribed quantity of interest. The efficiency of the proposed approach, which is particularly important if the number of independent random variables is large, is illustrated in 1D and 2D mechanical examples.