We theoretically propose a nonplasmonic optical refractive index sensor based on black phosphorus (BP) and other dielectric materials in the infrared band. Due to the anisotropic property of BP, the proposed sensor can achieve alternative sensitivity and figure of merit (FOM) in its different crystal directions. The high sensitivity and FOM are attributed to the strong magnetic resonance in the all-dielectric configuration. The coupled-mode theory (CMT) is used to verify the simulation results and reveal the physical mechanism. Furthermore, influences of the sample and the incident angle on the performance of the sensor are also discussed. Our design utilizes a simple dielectric structure with a BP monolayer, which exhibits great potential for the future high-performance sensor with low cost.