We investigate the precession of electron spins during beam-driven plasma-wakefield acceleration based on density down-ramp injection by means of full three-dimensional (3D) particle-in-cell (PIC) simulations. A relativistic electron beam generated via, e.g., laser wakefield acceleration, serves as the driving source. It traverses the prepolarized gas target and accelerates polarized electrons via the excited wakefield. We derive the criteria for the driving beam parameters and the limitation on the injected beam flux to preserve a high degree of polarization for the accelerated electrons, which are confirmed by our 3D PIC simulations and single-particle modeling. The electron-beam driver is free of the prepulse issue associated with a laser driver, thus eliminating possible depolarization of the prepolarized gas due to ionization by the prepulse. These results provide guidance for future experiments towards generating a source of polarized electrons based on wakefield acceleration.