We investigate the star formation properties of dynamically relaxed galaxy clusters as a function of cluster mass for 308 low-redshift clusters drawn from the Sloan Digital Sky Survey (SDSS) C4 cluster catalog. It is important to establish if cluster star formation properties have a mass dependence before comparing clusters at different epochs, and here we use cluster velocity dispersion, σ, as a measure of cluster mass. We select clusters with no significant substructure, a subset of the full C4 sample, so that velocity dispersion is an accurate tracer of cluster mass. We find that the total stellar mass, the number of star-forming galaxies, and total star formation rate scale linearly with the number of member galaxies, with no residual dependence on cluster velocity dispersion. With the mass-dependence of cluster star formation rates established, we compare the SDSS clusters with a sample of z 0.75 clusters from the literature and find that on average (correcting for the mass growth of clusters between the two redshifts) the total Hα luminosity of the high-redshift clusters is 10 times greater than that of the low-redshift clusters. This can be explained by a decline in the Hα luminosities of individual cluster galaxies by a factor of up to ~10 since z 0.75. The magnitude of this evolution is comparable to that of field galaxies over a similar redshift interval, and thus the effect of the cluster environment on the evolution of star-forming galaxies is at most modest. Our results suggest that the physical mechanism driving the evolution of cluster star formation rates is independent of cluster mass, at least for clusters with velocity dispersion greater than 450 km s−1, and operates over a fairly long timescale such that the star formation rates of individual galaxies decline by an order of magnitude over ~7 billion years.