The van der Waals heterojunctions of 2D materials offer tremendous opportunities in designing and investigating multifunctional and high-performance electronic and optoelectronic devices. In this study, a vertical p-n diode is constructed by vertically stacking p-type few-layer black phosphorus (BP) on n-type few-layer indium selenide (InSe). The photodetector based on the heterojunction displays a broadband and gate-modulated photoresponse under illumination. More importantly, by taking advantage of the strong linear dichroism of BP, the device demonstrates a highly polarization-sensitive photocurrent with an anisotropy ratio as high as 0.83. Additionally, the device can function in a zero-bias photovoltaic mode, enabling a fast photoresponse and low dark current. The external quantum efficiency can reach ≈3%, which is impressive for BP-based devices. The results pave the way for the implementation of p-BP/n-InSe heterostructure as a promising candidate for future multifunctional optoelectronics and, especially, polarization-sensitive applications.