Electron-capture reactions play important roles in the late evolution of core-collapse supernovae. The electron-capture rates used in astrophysical simulations rely on theoretical calculations which have to be tested against and guided by experimental data. We report on the measurement of the Gamow-Teller strength distribution of the odd-mass nucleus Nb93 via the (t,He3 + γ) charge-exchange reaction at a beam energy of 115 MeV/u. The Gamow-Teller strength distributions were extracted up to an excitation energy in Zr93 of 10 MeV. The results were compared with shell-model and quasiparticle random-phase approximation (QRPA) calculations. The theoretical calculations fail to describe the details of the strength distribution, but estimate reasonably well the integrated Gamow-Teller transition strength. Electron-capture rates derived from the measured and theoretical strength distributions match reasonably well, especially at the higher stellar densities of importance for deleptonization during the collapse of the stellar core, since the electron-capture Q value is close to zero and the Fermi energy sufficiently high to ensure that the details of the strength distribution do not have a strong impact on the derived rates. At stellar densities in excess of 109 g/cm3, the electron-capture rate based on a single-state approximation used in astrophysical simulations is slightly higher than the rates based on the data and the shell-model and QRPA calculations, likely due to the fact that the approximation includes temperature-dependent effects, which increase the rates. However, the difference is much smaller than that observed in recent studies of nuclei with Z<40 near N=50, suggesting that the single-state approximation does not account for Pauli-blocking effects for nuclei with Z<40 that are much stronger than those for Nb93 with Z=41.