In this paper we propose and analyze a finite element scheme for a class of variational nonlinear and nondifferentiable mixed inequalities including balance equations governing incompressible creeping flows of Bingham fluids. For numerical efficiency reasons, equal-order piecewise linear approximations are used for both velocity and pressure, and the numerical scheme is stabilized by a Brezzi-Pitkäranta perturbation term. We obtain error estimates of the same order as for stable discretizations, namely h1/2 for velocity and pressure solutions in \HtwoD and \Hun, respectively. A decomposition-coordination algorithm to solve the discrete nonlinear algebraic system is presented, together with its convergence properties. Finally, numerical tests are performed. The solution of the problem under consideration presents particular regularity properties that are shown to permit convergence order improvement to h |log(h)|1/2. This estimate is confirmed by numerical results.