In this paper, we investigate an reconfigurable intelligent surface (RIS)-aided integrated sensing and communication (ISAC) system. Our objective is to maximize the achievable sum rate of the multi-antenna communication users through the joint active and passive beamforming. {Specifically}, the weighted minimum mean-square error (WMMSE) method is { first} used to reformulate the original problem into an equivalent one. Then, we utilize an alternating optimization (AO) { algorithm} to decouple the optimization variables and decompose this challenging problem into two subproblems. Given reflecting coefficients, a penalty-based algorithm is utilized to deal with the non-convex radar signal-to-noise ratio (SNR) constraints. For the given beamforming matrix of the BS, we apply majorization-minimization (MM) to transform the problem into a quadratic constraint quadratic programming (QCQP) problem, which is ultimately solved using a semidefinite relaxation (SDR)-based algorithm. Simulation results illustrate the advantage of deploying RIS in the considered multi-user MIMO (MU-MIMO) ISAC systems.