We have undertaken a deep (σ ~ 1.1 mJy) 1.1-mm survey of the z = 0.54 cluster MS0451.6-0305 using the AzTEC camera on the James Clerk Maxwell Telescope. We detect 36 sources with signal-to-noise ratio (S/N) >= 3.5 in the central 0.10 deg2 and present the AzTEC map, catalogue and number counts. We identify counterparts to 18 sources (50 per cent) using radio, mid-infrared, Spitzer InfraRed Array Camera (IRAC) and Submillimetre Array data. Optical, near- and mid-infrared spectral energy distributions are compiled for the 14 of these galaxies with detectable counterparts, which are expected to contain all likely cluster members. We then use photometric redshifts and colour selection to separate background galaxies from potential cluster members and test the reliability of this technique using archival observations of submillimetre galaxies. We find two potential MS0451-03 members, which, if they are both cluster galaxies, have a total star formation rate (SFR) of ~100Msolaryr-1 - a significant fraction of the combined SFR of all the other galaxies in MS0451-03. We also examine the stacked rest-frame mid-infrared, millimetre and radio emission of cluster members below our AzTEC detection limit, and find that the SFRs of mid-IR-selected galaxies in the cluster and redshift-matched field populations are comparable. In contrast, the average SFR of the morphologically classified late-type cluster population is nearly three times less than the corresponding redshift-matched field galaxies. This suggests that these galaxies may be in the process of being transformed on the red sequence by the cluster environment. Our survey demonstrates that although the environment of MS0451-03 appears to suppress star formation in late-type galaxies, it can support active, dust-obscured mid-IR galaxies and potentially millimetre-detected LIRGs.