The transmission of terahertz (THz) Bessel vortex beams through a multi-layered anisotropic magnetized plasma slab is investigated by using a hybrid method of cylindrical vector wave functions (CVWFs) and Fourier transform. On the basis of the electromagnetic boundary conditions on each interface, a cascade form of expansion coefficients of the reflected and transmitted fields is obtained. Taking a double Gaussian distribution of the plasma density as an example, the influences of the applied magnetic field, the incident angle and polarization mode of the incident beams on the magnitude, OAM mode and polarization of the transmitted beams are analyzed in detail. The results indicate that the applied magnetic field has a major effect upon the polarization state of the transmitted fields but not upon the transmitted OAM spectrum. The incident angle has a powerful influence upon both the amplitude profile and the OAM spectrum of the transmitted beam. Furthermore, for multiple coaxial vortex beams, an increase of the maximum value of the plasma density causes more remarkable distortion of both the profile and OAM spectrum of the transmitted beam. This research makes a stable foundation for the THz OAM multiplexing/demultiplexing technology in magnetized plasma environment.