  We study the effect of the MeV-scale asymmetric dark matter annihilation on
the effective number of neutrinos $N_{\rm eff}$ at the epoch of the big bang
nucleosynthesis. If the asymmetric dark matter $\chi$ couples more strongly to
the neutrinos $\nu$ than to the photons $\gamma$ and electrons $e^-$,
$\Gamma_{\chi\gamma, \chi e} \ll \Gamma_{\chi\nu}$, or $\Gamma_{\chi\gamma,
\chi e} \gg \Gamma_{\chi\nu}$, the lower mass limit on the asymmetric dark
matter is about $18$ MeV for $N_{\rm eff}\simeq 3.0$.
