The search for neutrino events in correlation with gravitational wave (GW) events for three observing runs (O1, O2 and O3) from 09/2015 to 03/2020 has been performed using the Borexino data-set of the same period. We have searched for signals of neutrino-electron scattering and inverse beta-decay (IBD) within a time window of ±1000 s centered at the detection moment of a particular GW event. The search was done with three visible energy thresholds of 0.25, 0.8 and 3.0 MeV. Two types of incoming neutrino spectra were considered: the mono-energetic line and the supernova-like spectrum. GW candidates originated by merging binaries of black holes (BHBH), neutron stars (NSNS) and neutron star and black hole (NSBH) were analyzed separately. Additionally, the subset of most intensive BHBH mergers at closer distances and with larger radiative mass than the rest was considered. In total, follow-ups of 74 out of 93 gravitational waves reported in the GWTC-3 catalog were analyzed and no statistically significant excess over the background was observed. As a result, the strongest upper limits on GW-associated neutrino and antineutrino fluences for all flavors (ν_e, ν_μ, ν_τ) at the level 109-1015cm-2GW-1 have been obtained in the 0.5–5 MeV neutrino energy range.