The performance of the supplemental clutched inerter damper (CID) for the base-isolated multi-story structures subjected to near-fault earthquakes is investigated. The isolation system is considered as lead-rubber bearings with bi-linear characteristics and viscous damping. The resisting force of the CID is proportional to the relative acceleration between two terminals under the attached condition and zero when detached. The governing equations of motion of base-isolated structure and the CID are derived and solved using numerical technique under seven near-fault ground motions data. The variation of peak bearing displacement, top floor absolute acceleration, total base shear, and the CID force is plotted against the inertance mass ratio of the CID. The above peak responses were also analyzed for different values of damping, period of isolation, yield strength of LRB, and superstructure stories. Application of the CID is observed to effectively facilitate the reduction in bearing displacement while the combined effect of isolation and the CID prevents the top floor acceleration to shoot up. The optimum value of inertance mass ratio is also determined by minimizing the total base shear which is the measure of equivalent lateral force on the structure. The optimum inertance lies in the range of 35-45% of the total mass of the isolated building under near-fault motions. In addition, the performance of the CID base-isolated structure subjected to cycloidal pulses is also investigated. It is observed that the CID is quite effective in controlling the displacement of the isolation system under cycloidal pulses.