Thanks for raising the issue with supersymmetry (I work on supersymmetry from the exact moment I entered the field of particle physics). Supersymmetry (and its minimal incarnation in particular, as this is often what is referred to when supersymmetry is mentioned) is a very popular framework for physics beyond the Standard Model, although other options are getting more and more traction (by virtue of no signal of supersymmetry in data).
However, for what concerns neutrino masses minimal supersymmetry does not help. We indeed need to go beyond the minimal framework so that we have chances to have non-zero neutrino masses. I have personally worked on some interesting models in which we supersymmetrise the Standard Model and generalise in addition the structure of the fundamental interaction. The former allows for all the advantages of a supersymmetric theory, whereas the latter offers a perfect framework for modelling neutrino masses and mixings.
Cheers, and thanks again for this nice and interesting comment.