The search for neutrino events in correlation with 42 most intense fast radio bursts (FRBs) has been performed using the Borexino dataset from 05/2007 to 06/2021. We have searched for signals with visible energies above 250 keV within a time window of ±1000 s corresponding to detection time of a particular FRB. We also applied an alternative approach based on searching for specific shapes of neutrino-electron scattering spectra in the full exposure data of the Borexino detector. In particular, two incoming neutrino spectra were considered: the monoenergetic line and the spectrum expected from supernovae. The same spectra were considered for electron antineutrinos detected through inverse beta-decay reaction. No statistically significant excess over the background was observed. As a result, the strongest upper limits on FRB-associated neutrino fluences of all flavors have been obtained in the 0.5–50 MeV neutrino energy range.