In this paper, we study the sensitivity-tunable terahertz (THz) liquid/gas biosensor in a coupling prism–three-dimensional Dirac semimetal (3D DSM) multilayer structure. The high sensitivity of the biosensor originates from the sharp reflected peak caused by surface plasmon resonance (SPR) mode. This structure achieves the tunability of sensitivity due to the fact that the reflectance could be modulated by the Fermi energy of 3D DSM. Besides, it is found that the sensitivity curve depends heavily on the structural parameters of 3D DSM. After parameter optimization, we obtained sensitivity over 100°/RIU for liquid biosensor. We believe this simple structure provides a reference idea for realizing high sensitivity and a tunable biosensor device.