This work is devoted to study efficient numerical methods for solving nonsmooth convex equilibrium problems in the sense of Blum and Oettli. First we consider the auxiliary problem principle which is a generalization to equilibrium problems of the classical proximal point method for solving convex minimization problems. This method is based on a fixed point property. To make the algorithm implementable we introduce the concept of $mu$-approximation and we prove that the convergence of the algorithm is preserved when in the subproblems the nonsmooth convex functions are replaced by $mu$-approximations. Then we explain how to construct $mu$-approximations using the bundle concept and we report some numerical results to show the efficiency of the algorithm. In a second part, we suggest to use a barrier function method for solving the subproblems of the previous method. We obtain an interior proximal point algorithm that we apply first for solving nonsmooth convex minimization problems and then for solving equilibrium problems. In particular, two interior extragradient algorithms are studied and compared on some test problems.