Based on density functional theory calculation, herein we propose $X$MoSiP$_2$ ($X=$ S, Se, Te) monolayers for new two-dimensional (2D) Janus materials. Their crystal structures with dynamical, mechanical, and thermal stabilities, electronic and transport properties are systematically investigated. The results reveal that all three $X$MoSiP$_2$ monolayers exhibit isotropic elastic properties with high Young's modulus and negative cohesive energy values, as well as the elastic constants follow the Born-Huang's criteria, demonstrating their mechanical stability and suggesting the high ability for the experimental synthesis of these materials. From the Perdew-Burke-Ernzerhof (PBE) functional, the SMoSiP$_2$ is observed as a semiconductor with an indirect bandgap of 1.01 eV, while the SeMoSiP$_2$ and TeMoSiP$_2$ monolayers are observed as direct semiconductors with the bandgap energy of 1.09 eV and 1.12 eV, respectively. Notably, the bandgap energy of the materials is changed significantly by applying the biaxial strain, and the transition between direct and indirect semiconductors is observed. For the transport ability of the materials, the carrier mobilities are found to be anisotropic for both electron and hole in our studied materials. These finding further highlights the extraordinary properties of the 2D Janus $X$MoSiP$_2$ materials and their promise to apply in electronic devices.