We have conducted a survey of young single and multiple systems in the Taurus-Auriga star-forming region with the Atacama Large Millimeter Array (ALMA), substantially improving both the spatial resolution and sensitivity with which individual protoplanetary disks in these systems have been observed. These ALMA observations can resolve binary separations as small as 25-30au and have an average 3{sigma} detection level of 0.35mJy, equivalent to a disk mass of 4x10^-5^M_{sun}_ for an M3 star. Our sample was constructed from stars that have an infrared excess and/or signs of accretion and have been classified as Class II. For the binary and higher-order multiple systems observed, we detect {lambda}=1.3mm continuum emission from one or more stars in all of our target systems. Combined with previous surveys of Taurus, our 21 new detections increase the fraction of millimeter-detected disks to over 75% in all categories of stars (singles, primaries, and companions) earlier than spectral type M6 in the Class II sample. Given the wealth of other information available for these stars, this has allowed us to study the impact of multiplicity with a much larger sample. While millimeter flux and disk mass are related to stellar mass as seen in previous studies, we find that both primary and secondary stars in binary systems with separations of 30-4200au have lower values of millimeter flux as a function of stellar mass than single stars. We also find that for these systems, the circumstellar disk around the primary star does not dominate the total disk mass in the system and contains on average 62% of the total mass.