We perform calculations of the cross sections for charged-current neutrino and antineutrino scattering off 116Cd using ten different Skyrme interactions, at energies typical of supernova neutrinos. We use the quasiparticle random-phase approximation in its charged-changing mode (pnQRPA) to construct the required nuclear wave functions for the participant initial and final states. We compare the results of these calculations with the results of calculations based on the Bonn one-boson-exchange potential. The response of 116Cd to supernova neutrinos is calculated by folding the obtained cross sections with suitably parametrized Fermi-Dirac distributions of the electron-neutrino and electron-antineutrino energies.