We investigate the role of dense environments in suppressing star formation by studying log(10)(M- * / M-?) > 9 . 7 star-forming galaxies in nine clusters from the Local Cluster Survey (0.0137 < z < 0.0433) and a large comparison field sample drawn from the Sloan Digital Sky Survey. We compare the star formation rate (SFR) with stellar mass relation as a function of environment and morphology. After carefully controlling for mass, we find that in all environments, the degree of SFR suppression increases with increasing bulge-to-total (B/T) ratio. In addition, the SFRs of cluster and infall galaxies at a fixed mass are more suppressed than their field counterparts at all values of B/T. These results suggest a quenching mechanism that is linked to bulge growth that operates in all environments and an additional mechanism that further reduces the SFRs of galaxies in dense environments. We limit the sample to B/T = 0.3 galaxies to control for the trends with morphology and find that the excess population of cluster galaxies with suppressed SFRs persists. We model the time-scale associated with the decline of SFRs in dense environments and find that the observed SFRs of the cluster core galaxies are consistent with a range of models including a mechanism that acts slowly and continuously o v er a long (2-5 Gyr) time-scale, and a more rapid ( < 1 Gyr) quenching event that occurs after a delay period of 1-6 Gyr. Quenching may therefore start immediately after galaxies enter clusters.