The process of v (v) over bar Bremsstrahlung via electron scattering off nuclei in the crust of a cooling neutron star is studied within the Random Phase Approximation. We show that the electromagnetic plasma screens the vector weak coupling of electrons with the neutrino field. This screening effect takes place in both the crystalline and liquid phase of the neutron star crust, when the momentum carried out by the neutrino pair is smaller than the inverse Debye screening distance of the electrons. This condition holds in the crust volume at temperatures and densities where the electron Bremsstrahlung dominates the neutrino production. As a result, the vector weak current contribution to the v (v) over bar emissivity decreases along with the temperature as T-10, contrarily to the axial term, which is proportional to T-6. At sufficiently low temperatures, typical for a crystalline crust, the total neutrino-pair emissivity is lower in approximately a factor 2 than previously estimated. We argue that the screening effects under consideration can influence the thermal radiation from the neutron star surface, which is potentially observable.