We report on strong plasmonic coupling from silver nanoparticles covered by hydrogen-terminated chemically vapor deposited single-layer graphene, and its effects on the detection and identification of adenine molecules through surface-enhanced Raman spectroscopy (SERS). The high resistivity of the graphene after subjecting to remote plasma hydrogenation allows plasmonic coupling induced strong local electromagnetic fields among the silver nanoparticles to penetrate the graphene, and thus enhances the SERS efficiency of adenine molecules adsorbed on the film. The graphene layer protects the nanoparticles from reactive and harsh environments and provides a chemically inert and biocompatible carbon surface for SERS applications.