We study the spectroscopic properties and environments of red (or passive) spiral galaxies found by the Galaxy Zoo project. By carefully selecting face-on disc-dominated spirals, we construct a sample of truly passive discs (i.e. they are not dust reddened spirals, nor are they dominated by old stellar populations in a bulge). As such, our red spirals represent an interesting set of possible transition objects between normal blue spiral galaxies and red early types, making up ∼6 per cent of late-type spirals. We use optical images and spectra from Sloan Digital Sky Survey to investigate the physical processes which could have turned these objects red without disturbing their morphology. We find red spirals preferentially in intermediate density regimes. However, there are no obvious correlations between red spiral properties and environment suggesting that environment alone is not sufficient to determine whether a galaxy will become a red spiral. Red spirals are a very small fraction of all spirals at low masses (M★ < 1010 M⊙), but are a significant fraction of the spiral population at large stellar masses showing that massive galaxies are red independent of morphology. We confirm that as expected, red spirals have older stellar populations and less recent star formation than the main spiral population. While the presence of spiral arms suggests that a major star formation could not have ceased a long ago (not more than a few Gyr), we show that these are also not recent post-starburst objects (having had no significant star formation in the last Gyr), so star formation must have ceased gradually. Intriguingly, red spirals are roughly four times as likely than the normal spiral population to host optically identified Seyfert/low-ionization nuclear emission region (LINER; at a given stellar mass and even accounting for low-luminosity lines hidden by star formation), with most of the difference coming from the objects with LINER-like emission. We also find a curiously large optical bar fraction in the red spirals (70 ± 5 verses 27 ± 5 per cent in blue spirals) suggesting that the cessation of star formation and bar instabilities in spirals are strongly correlated. We conclude by discussing the possible origins of these red spirals. We suggest that they may represent the very oldest spiral galaxies which have already used up their reserves of gas – probably aided by strangulation or starvation, and perhaps also by the effect of bar instabilities moving material around in the disc. We provide an online table listing our full sample of red spirals along with the normal/blue spirals used for comparison.