The introduction of high energy dissipation in base-isolated structures is often prescribed to minimize the device displacements as well as the effects of near-field earthquakes. The identification of effects on the superstructure due to the high energy dissipation is, therefore, an important aspect of the base-isolated structure design. In this study, the seismic response of base-isolated structures with Lead Rubber Bearing (LRB) devices is estimated aiming at the evaluation of the adverse effect of damping on the structural response parameters. Four base-isolated structures are considered taking into account a complete damping matrix. Their structural seismic response evaluation is first performed using nonlinear response history analysis (NRHA) by considering a bilinear device behaviour. The increase in the superstructure response parameters is detected. A structural analysis by considering an equivalent linear viscoelastic LRB behaviour was also performed. A frequency domain method through transmissibility was applied to explain the influence of isolation damping on the higher mode effects and inter-storey drift ratios. The comparison between the NRHA results and response spectrum analysis (RSA) results highlights meaningful differences between the values of some structural response parameters (displacements of the isolation system and inter-storey drift ratio). A seismic analysis of base-isolated structures with High Damping Rubber Bearing (HDRB) and supplemental linear viscous damping (VD) devices is also carried out. The results point out that the use of HDRB devices with linear viscous dampers, as compared to LRB devices, lead to a reduction of the devices displacements and to a beneficial or least detrimental effects on the superstructure response parameters in base-isolated structures.