The energy of the neutrino matters more than its rest mass. Consider the reaction
##\bar \nu_e + p \to e^+ + n##
that is: an electron-antineutrino hits a proton and in the process of the collision the proton turns into a neutron and the antineutrino turns into a positron. This happens via the exchange of a W boson between the antineutrino and the proton.
You are right that a lepton like a positron is much heavier than a neutrino. Also, a neutron is heavier than a proton. The extra rest-mass energy is supplied by the kinetic energy of the initial particles. Therefore this reaction can only happen if the antineutrino has enough kinetic energy to supply the missing rest-mass energy needed to make the final-state particles.
The W+ boson is often a virtual particle - "its energy" (note: this is not really a well-defined value here) can be below the W mass.
To produce positrons or other reaction products, it needs enough kinetic energy.
The requirement for neutrinos to have extra kinetic energy (or be at a higher energy level) is one of the reasons they rarely experience a charged current of the weak force, despite there being so many of them in existence.