We describe the first optical survey of absorption systems associated with cluster galaxies at z = 0.3-0.9. We have cross-correlated quasars from the third data release of the SDSS with high-redshift cluster/group candidates from the Red-Sequence Cluster Survey. We have found 442 quasar-cluster pairs for which the Mg II lambda lambda 2796, 2803 doublet might be detected at a transverse (physical) distance d < 2 h(71)(-1) Mpc from the cluster centers. To investigate the incidence dN/dz and equivalent width distribution n(W) of Mg II systems at cluster redshifts, two statistical samples were drawn out of these pairs: one made of high-resolution spectroscopic quasar observations (46 pairs), and one made of quasars used in Mg II searches found in the literature (375 pairs). The results are (1) the population of strong Mg II systems (W-0(2796) > 2.0 angstrom) near cluster redshifts shows a significant (> 3 sigma) overabundance (up to a factor of 15) when compared with the "field'' population; (2) the overabundance is more evident at d < 1 h(71)(-1) Mpc than at d < 2 h(71)(-1) 71 Mpc, and more evident in a subsample of the most massive clusters; and ( 3) the population of weak Mg II systems (W-0(2796) < 0.3 angstrom) near cluster redshifts conforms to the field statistics. Unlike in the field, this dichotomy makes n( W) in clusters appear flat and well fitted by a power law in the entire Wrange. Since either the absorber number density or the filling factor/cross section affects the absorber statistics, an interesting possibility is that we have detected the signature of truncated halos due to environmental effects. Thus, the excess of strong systems is due to a population of absorbers in an overdense galaxy region, and the lack of weak systems to a different population, that got destroyed in the cluster environment.