Effects of neutrino charge radius and magnetic moment, as well as the medium modifications of the weak and electromagnetic nucleon form factors of the constituents of matter on the neutrino electroweak interaction with dense nuclear matter, are estimated. A relativistic mean-field and quark-meson coupling models are adopted for the in-medium effective nucleon mass and nucleon form factors. We find that the neutrino scattering cross section increases in the cold nuclear medium when neutrino form factors and the in-medium modifications of the nucleon weak and electromagnetic form factors are simultaneously taken into account relative to that without neutrino form factors. The increase of the cross section results in the decrease of the neutrino mean free path, particularly at larger neutrino magnetic moment and charge radius. The quenching of the neutrino mean free path is estimated to be about 12-58% for the values of $mu_ u = 3 times 10^{-12} mu_B$ and $R_ u = 3.5 times 10^{-5}~textrm{MeV}^{-1}$, obtained from the constraints of the astrophysical observations, compared to that of $mu_ u =0$ and $R_ u =0$. The decrease of the neutrino mean free path is expected to decelerate the cooling of neutron stars. Each contribution of the neutrino form factors to the neutrino mean free path is discussed.
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