For the hydromechanical coupling of solid-fluid porous media, this study presents an explicit stabilized two-phase material point method (MPM) formulation based on the one-point two-phase MPM scheme. To mitigate the spurious pore pressure and maintain the numerical stability, stabilized techniques, including the strain smoothing method and the multi-field variational principle, are implemented in the proposed formulation. The strain smoothing technique is used to smooth the volumetric strain rate, and the calculation of the pore pressure increment at particles is based on the multi-field variational principle. Four numerical examples are performed to evaluate the performance of the proposed formulation. With its effective and easy-to-implement stabilized techniques, the proposed formulation provides stable and reliable outcomes that align well with analytical solutions and results from other approaches, offering extensive validation that the proposed two-phase MPM formulation is an effective and reliable approach for the simulation of solid-fluid porous media under both static and dynamic conditions.