We present algebraic multilevel preconditioners for linear systems arising from the discretization of systems of coupled elliptic partial differential equations (PDEs). These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods; in particular, when applied to the optimality systems associated with elliptic PDE-constrained optimization problems, these preconditioners appear to be rather insensitive to the regularization parameter in the cost functional. A convergence theory for the twolevel case is presented.