Two-dimensional (2D) semiconductors have attracted a great deal of interest from the electrical engineering community due to their intriguing electronic properties. In this study, we have systematically investigated the electronic and optoelectronic properties of β-AsP monolayers by first-principles calculations combined with strain engineering. The results show that the β-AsP monolayer has a suitable indirect bandgap and a strain-tunable electronic structure. On this basis, the designed two-electrode photodetector based on β-AsP monolayer exhibits a strong photoelectric response in the near-ultraviolet region, and the maximum photocurrent can reach 74a02 per phonon under the applied bias voltage of 1 V. In particular, the strain engineering not only improves the photoelectric performance of the β-AsP-based photodetector, but also adjusts its photodetection range. These findings suggest that β-AsP monolayers are ideal candidates for the development of near-ultraviolet photodetectors and optoelectronic devices.