We present a high angular resolution millimeter-wave dust continuum imaging survey of circumstellar material associated with the individual components of 23 multiple star systems in the Taurus-Auriga young cluster. Combined with previous measurements in the literature, these new data permit a comprehensive look at how the millimeter luminosity (a rough tracer of disk mass) relates to the separation and mass of a stellar companion. Approximately one-third (28%-37%) of the individual stars in multiple systems have detectable millimeter emission, an incidence rate half that for single stars (~62%) which does not depend on the number of companions. There is a strong, positive correlation between the luminosity and projected separation (ap ) of a stellar pair. Wide pairs (ap > 300 AU) have a similar luminosity distribution as single stars, medium pairs (ap ≈ 30-300 AU) are a factor of five fainter, and close pairs (ap < 30 AU) are ~5× fainter yet (aside from a small, but notable population of bright circumbinary disks). In most cases, the emission is dominated by a disk around the primary (or a wide tertiary in hierarchical triples), but there is no clear relationship between luminosity and stellar mass ratio. A direct comparison of resolved disk sizes with predictions from tidal truncation models yields mixed results; some disks are much larger than expected given the projected distances of their companions. We suggest that the presence of a stellar companion impacts disk properties at a level comparable to the internal evolution mechanisms that operate in an isolated system, with both the multiple star formation process itself and star-disk tidal interactions likely playing important roles in the evolution of circumstellar material. From the perspective of the mass content of the disk reservoir, we expect that (giant) planet formation is inhibited around the components of close pairs or secondaries, but should be as likely as for single stars around the primaries (or wide tertiaries in hierarchical triples) in more widely separated multiple star systems.
