The Rashba spin-splitting and thermoelectric transport of two-dimensional black arsenic phosphorus (2D B-AsP) under strain control are studied using non-equilibrium Green’s function and first principles calculations. Firstly, we determine the stability range of 2D B-AsP under strain conditions, with biaxial strain ranging from −2% to +5% and uniaxial strain ranging from −4% to +5%. Secondly, we analyze at the point Γ under strain within the range of −1% to +2%, and find the spin-splitting coefficient decreases with increasing strain. Thirdly, we discuss the effects of applying strain and changing temperature on 2D B-AsP thermoelectric transport. When the uniaxial compression strain is −4%, the ZT coefficient increases by 12.91%, and increases by 48.8% at a temperature of 400 K. Finally, when two external conditions exist simultaneously, the ZT coefficient of the material increases by 44.6%. The research results demonstrate the potential applications of 2D B-AsP in spintronics and thermoelectricity.