The neutrino-nucleus reaction cross sections of ^4^He and ^12^C are evaluated using new shell model Hamiltonians. Branching ratios of various decay channels are calculated to evaluate the yields of Li, Be, and B produced through the {nu}-process in supernova explosions. The new cross sections enhance the yields of ^7^Li and ^11^B produced during the supernova explosion of a 16.2M_{sun} star model compared to the case using the conventional cross sections by about 10%. On the other hand, the yield of ^10^B decreases by a factor of 2. The yields of ^6^Li, ^9^Be, and the radioactive nucleus ^10^Be are found at a level of ~10^-11^M{sun}. The temperature of {nu}{mu},{tau} - and {nu}{bar}{mu},{tau}_ - neutrinos inferred from the supernova contribution of ^11^B in Galactic chemical evolution models is constrained to the 4.3-6.5MeV range. The increase in the ^7^Li and ^11^B yields due to neutrino oscillations is demonstrated with the new cross sections.