We consider the role of Majoron emission in supernova cooling and its implications for the neutrino mass and lifetime in generic single Majoron models. It is found that, for ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ with mass $m$, if the lifetime for the decay ${\ensuremath{\nu}}_{\ensuremath{\tau}}\ensuremath{\rightarrow}\mathrm{Majoron}+{\ensuremath{\nu}}_{e,\ensuremath{\mu}}$ is shorter than ${10}^{\ensuremath{-}7}$ (m/MeV) sec, then Majorons are so strongly trapped by the inverse process that the resulting Majoron luminosity is small enough not to destabilize the observed ${\ensuremath{\nu}}_{e}$ pulse from SN 1987A. For ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ with a longer lifetime, the Majoron luminosity can be large enough to destroy or significantly shorten the duration of the neutrino pulse. We then find the range of parameters, e.g., the ${\ensuremath{\nu}}_{\ensuremath{\tau}}$ mass $m$ and the $B\ensuremath{-}L$-breaking scale $v$, that is excluded by giving such a large Majoron luminosity. Our results imply that, for $v$ between 1 GeV and 1 TeV, a wide range of $m$ allowed by terrestrial experiments can be excluded in view of the observed ${\ensuremath{\nu}}_{e}$, pulse from SN 1987A.