Two-dimensional (2D) halide perovskites have demonstrated impressive long-term stability and superior device performance as compared to their three-dimensional (3D) counterparts. The potential of 2D halide perovskites for advanced photovoltaic applications can be enhanced by an understanding of how external factors like strain could be used to tune their optoelectronic properties. This study explores the effects of biaxial strain on the structure and electronic transport properties of 2D halide perovskites, focusing on the lowest energy (001) surfaces of (Cs2BCl4 and CsB2Cl5, B = Pb or Sn) with CsCl and BCl2 terminations. Using first-principles calculations, we find that the lower energy CsCl terminated surface, resulting in Cs2BCl4, couples strongly with biaxial strain. This termination shows bandgap modulations from approximately 1.5 eV to 1.8 eV for Cs2PbCl4 and 1.2 eV to 1.5 eV for Cs2SnCl4 with biaxial strain. Within the acoustic deformation potential theory, we compute hole mobilities, and find substantial enhancements of approximately 80% for Pb-based and 50% for Sn-based systems, thereby emphasizing the potential of strain engineering to further optimize charge transport properties in 2D halide perovskites.