GW170817 and electromagnetic follow-up observations showed that binary neutron star mergers are accompanied by short gamma-ray bursts (sGRBs) and kilonovae.The former and the latter are produced by jets and ejecta from neutron stars. sGRBs have extended emission components lasting 100-1000 s, whose emission mechanism is under debate. The jet emissions produced by prolonged central engine activity could explain the components. The jet should propagate inside the ejecta, forming a cocoon around the jet. Photons filled in the cocoon should diffuse into the dissipation region of the jet. We calculate the neutrino emission from the jet due to prolonged engine activity, taking into account the interaction between cocoon photons and cosmic rays accelerated in jets. We find that the detection of neutrino signals associated with gravitational waves is achieved with a high probability for 10 years of operation by the future project, IceCube-Gen2, and the second-generation gravitational wave detectors. Otherwise, we can put meaningful constraints on dissipation radius and baryon loading factor.