The first-passage time probability plays an important role in the reliability assessment of dynamic systems in random vibrations. To find the solution of the first-passage time probability is a challenging task. The analytical solution to this problem is not available even for linear dynamic systems. For nonlinear and multi-degree-of-freedom systems, it is even more challenging. This paper proposes a radial basis function neural networks method for solving the first-passage time probability problem of linear, nonlinear, and multi-degree-of-freedom dynamic systems. In this paper, the proposed method is applied to solve for the backward Kolmogorov equation subject to boundary conditions defined by the safe domain. A null-space solution strategy is proposed to deal with the boundary condition. Several examples including a two degrees-of-freedom nonlinear Duffing system are studied with the proposed method. The results are compared with Monte Carlo simulations. It is believed that the radial basis function neural networks method provides a new and effective tool for the reliability assessment and design of multi-degree-of-freedom nonlinear stochastic dynamic systems.