Protoneutron stars formed during core-collapse supernovae are hot and dense environments that contain a sizable population of muons. If these interact with new long-lived particles with masses up to roughly 100 MeV, the latter can be produced and escape from the stellar plasma, causing an excessive energy loss constrained by observations of SN 1987A. In this article we calculate the emission of light dark fermions that are coupled to leptons via a new massive vector boson, and determine the resulting constraints on the general parameter space. We apply these limits to the gauged Lμ-Lτ model with dark fermions, and show that the SN 1987A constraints exclude a significant portion of the parameter space targeted by future experiments. We also extend our analysis to generic effective four-fermion operators that couple dark fermions to muons, electrons, or neutrinos. We find that SN 1987A cooling probes a new physics scale up to ∼7 TeV , which is an order of magnitude larger than current bounds from laboratory experiments.