The paper presents an algorithm for solving minimax problems of stochastic programming: \[ \min_{x\in X}\max_{H\in G}\int_ \Omega f(x,\omega)dH(\omega) \] with \(f(.,\omega)\) convex, \(X\) convex closed and \(G\) a given set of probability measures defined by means of finitely many moment conditions. The algorithm is based on the stochastic quasigradient method combined with an exploitation of sample information and it opens the possibility to solve numerically the minimax problem without restrictive assumptions concerning the set \(G\). Convergence is proved and numerical experiments are reported. A special version of the algorithm is formulated for solving minimax stochastic linear programs with complete recourse.