Core Collapse Supernovae (CCSN) are explosive phenomena that may occur at the end of the life of massive stars, releasing over 99% of the energy through neutrino emission with energies on the 10 MeV scale. While the explosion mechanism is not fully understood, neutrinos are believed to play an important role. The only detection as of today are the 24 neutrinos from supernova SN1987A. The observation of the next Galactic CCSN will lead to important breakthroughs across the fields of astrophysics, nuclear and particle physics. For a Galactic CCSN, the KM3NeT ORCA and ARCA detectors in the Mediterranean Sea will observe a significant number of neutrinos via the detection of Cherenkov light, mostly induced by Inverse Beta Decay (IBD) interactions in sea water. The detection of coincident photons by the 31 photomultipliers of the KM3NeT digital optical modules (DOMs) allows to separate the signal from the optical background sources. The KM3NeT detection sensitivity to a Galactic CCSN and the potential to resolve the neutrino light-curve have been estimated exploiting detailed Monte-Carlo simulations. Specific criteria are proposed for the online triggering and the participation in the SNEWS network.