In this paper we are concerned with a linear Boussinesq system of Benjamin–Bona–Mahony type modelling the two-way propagation of surface waves in a uniform horizontal channel filled with an irrotational, incompressible and inviscid liquid under the influence of gravitation. We propose several dissipation mechanisms leading to systems for which one has both the global existence of solutions and a nonincreasing energy. Following the analysis developed in Rosier (J Math Study 49:195–204, 2016) we prove that all the trajectories are attracted by the origin provided that the unique continuation of weak solutions holds.