Context: Open clusters (OCs) are widely used as test particles to investigate a variety of astrophysical phenomena, from stellar evolution to Galactic evolution. Gaia and the complementary massive spectroscopic surveys are providing an unprecedented wealth of information about these systems. Aims: The Open Cluster Chemical Abundances from Spanish Observatories (OCCASO) survey aims to complement all this work by determining OCs accurate radial velocities and chemical abundances from high-resolution, R$\geq$60\,000, spectra. Methods: Radial velocities have been obtained by cross-correlating the observed spectra with a library of synthetic spectra which covers from early M to A spectral types. Results: We provide radial velocities for 336 stars, including several Gaia Benchmark Stars and objects belonging to 51 open clusters. The internal uncertainties of the derived radial velocities go from 10 m/s to 21 m/s as a function of the instrumental configuration used. The derived radial velocities, together with the Gaia proper motions, have been used to investigate the cluster membership of the observed stars. After this careful membership analysis, we obtain average velocities for 47 open clusters. To our knowledge, this is the first radial velocity determination for 5 of these clusters. Finally, the radial velocities, proper motions, distances and ages have been used to investigate the kinematics of the observed clusters and in the integration of their orbits.