The traditional base-isolated system is vulnerable to long-period ground motions, which usually result in a large displacement concentration at the isolated floor due to the resonant effect. To address this issue, two types of base isolation systems with tuned inerter dampers (TID) composed of a spring, an inerter and a dashpot in serial or parallel, are proposed and evaluated in this paper. The design parameters of the two TID isolation systems are optimized using the H2 norm criteria to achieve the best RMS vibration performance under stochastic excitation. The TID frequency ratio and damping ratio are defined as the design parameters, whose optimal values are analytically derived for the undamped primary system and numerically verified. The results show that the optimum exists for isolation system with serial TID (inerter and dashpot in serious), while in the parallel TID isolation system large TID stiffness and large TID damping are preferred in practice. The parallel TID system cannot be tuned optimally for practical structures, nevertheless, it still achieves a better isolation performance than the optimal serial system by an appropriate selection of the design parameters. The influence of the structural parameters on the optimal design parameters are studied. Case studies are conducted in comparison with the traditional isolation system for a laboratory prototype of a five-story building. The proposed optimal serial TID isolation system has 59% more reduction in the RMS relative displacement between the superstructure and base and 58% in the RMS response of the base vibration under the far-fault earthquake. And 52% and 56% more reductions in the RMS relative displacement and the base vibration are respectively achieved under the near-fault earthquake. The potential power in the TID isolations in earthquakes are also examined.