The shear viscosity coefficients of water and water-based nanofluids with copper particles are calculated by the molecular dynamics method. Copper nanoparticles with a diameter of 2, 4 and 10 nm were used in the simulation. The volume fraction of nanoparticles was varied from 1 to 5%. The interaction of water molecules with each other was modeled using the Lennard–Jones potential. The Rudyak–Krasnolutskii and Rudyak–Krasnolutskii–Ivanov potentials were used as nanoparticle–molecule and nanoparticles interaction potentials, respectively. The viscosity coefficient was calculated using the fluctuation–dissipation theorem by the Green–Kubo formula. It is shown that the viscosity of the nanofluid significantly exceeds the viscosity of the coarse-grained suspension and increases with a decrease in the nanoparticles size at their fixed volume fraction. The correlation functions determining the viscosity coefficient of the nanofluid were analyzed in detail. The radial distribution functions of pure water and nanofluids are also presented in the paper. It is shown that the liquid near the nanoparticle is structured much more strongly than in the bulk. This greater ordering of the nanofluid is one of the main factors determining the increase in nanofluids viscosity.