We have studied the radio and far-infrared (FIR) emission from 114 galaxies in the seven nearest clusters (<100 Mpc) with prominent X-ray emission to investigate the impact of the cluster environment on the star formation and active galactic nucleus (AGN) activity in the member galaxies. The X-ray selection criterion is adopted to focus on the most massive and dynamically relaxed clusters. A large majority of cluster galaxies show an excess in radio emission over that predicted from the radio-FIR correlation, the fraction of sources with radio excess increases toward cluster cores, and the radial gradient in the FIR/radio flux ratio is a result of radio enhancement. Of the radio-excess sources, 70% are early-type galaxies, and the same fraction host an AGN. The galaxy density drops by a factor of 10 from the composite cluster center out to 1.5 Mpc, yet galaxies show no change in FIR properties over this region and show no indication of mass segregation. We have examined in detail the physical mechanisms that might impact the FIR and radio emission of cluster galaxies. While collisional heating of dust may be important for galaxies in cluster centers, it appears to have a negligible effect on the observed FIR emission for our sample galaxies. The correlations between radio and FIR luminosity and radius could be explained by magnetic compression from thermal intracluster medium pressure. We also find that simple delayed harassment cannot fully account for the observed radio, FIR, and mid-IR properties of cluster galaxies.