Swift electrons moving closely parallel to a periodic grating produce far-field radiation of light, which is known as the Smith-Purcell effect. In this letter, we demonstrate that designer Babinet metasurfaces composed of C-aperture resonators offer a powerful control over the polarization state of the Smith-Purcell emission, which can hardly be achieved via traditional gratings. By coupling the intrinsically nonradiative energy bound at the source current sheet to the out-of-plane electric dipole and in-plane magnetic dipole of the C-aperture resonator, we are able to excite cross-polarized light thanks to the bianisotropic nature of the metasurface. The polarization direction of the emitted light is aligned with the orientation of the C-aperture resonator. Furthermore, the efficiency of the Smith-Purcell emission from Babinet metasurfaces is significantly increased by 84%, in comparison with the case of conventional gratings. These findings not only open up a new way to manipulate the electron-beam-induced emission in the near-field region but also promise compact, tunable, and efficient light sources and particle detectors.