Damping enhancement is the typical characteristic of the inerter system for seismic response mitigation. To give full play to this characteristic and meet the demand of seismic performance at the same time，it is proposed to maximize the degree of damping enhancement of the inerter system with the performance demand as the constraint condition during the decision of key parameters. The closed-form solution of a single-degree-of-freedom structure with an inerter system under the excitation of white noise is derived based on the theory of random vibration. The mathematical expression of the equivalent constrained optimization problem for damping enhancement maximization is established. Given the complexity of the expression，the particle swarm algorithm，a robust and simple meta-heuristics method for numerical optimization，is used to solve the problem. The adaptively changed penalty weight is introduced into the particle swarm algorithm to consider the constraints，and the adaptively adjusted inertia weight is used to improve the efficiency for the search of the optimal solution. A computer program of the adaptively weighted particle swarm algorithm is developed to solve the design problem of the inerter system for damping enhancement maximization. The design examples reflect the effectiveness of the adaptively weighted particle swarm algorithm in solving the optimization problem of the structure with inerter system. And the dynamic time-history analyses show that the structural damping performance demands are realized with the designed parameters.