We investigate the properties of optically passive spirals and dusty red galaxies in the A901/2 cluster complex at redshift similar to 0.17 using rest-frame near-ultraviolet-optical spectral energy distributions, 24-mu m infrared data and Hubble Space Telescope morphologies from the STAGES data set. The cluster sample is based on COMBO-17 redshifts with an rms precision of sigma(cz) approximate to 2000 km s(-1). We find that 'dusty red galaxies' and 'optically passive spirals' in A901/2 are largely the same phenomenon, and that they form stars at a substantial rate, which is only four times lower than that in blue spirals at fixed mass. This star formation is more obscured than in blue galaxies and its optical signatures are weak. They appear predominantly in the stellar mass range of log M-*/M-circle dot = [10, 11] where they constitute over half of the star-forming galaxies in the cluster; they are thus a vital ingredient for understanding the overall picture of star formation quenching in clusters. We find that the mean specific star formation rate (SFR) of star-forming galaxies in the cluster is clearly lower than in the field, in contrast to the specific SFR properties of blue galaxies alone, which appear similar in cluster and field. Such a rich red spiral population is best explained if quenching is a slow process and morphological transformation is delayed even more. At log M-*/M-circle dot < 10, such galaxies are rare, suggesting that their quenching is fast and accompanied by morphological change. We note that edge-on spirals play a minor role; despite being dust reddened they form only a small fraction of spirals independent of environment.