We conduct a multiplicity survey of members of the {rho} Ophiuchus cloud complex with high-resolution imaging to characterize the multiple-star population of this nearby star-forming region and investigate the relation between stellar multiplicity and star and planet formation. Our aperture masking survey reveals the presence of five new stellar companions beyond the reach of previous studies, but does not result in the detection of any new substellar companions. We find that 43+/-6% of the 114 stars in our survey have stellar-mass companions between 1.3 and 780 AU, while 7_-5_^+8^% host brown dwarf companions in the same interval. By combining this information with knowledge of disk-hosting stars, we show that the presence of a close binary companion (separation <40AU) significantly influences the lifetime of protoplanetary disks, a phenomenon previously seen in older star-forming regions. At the ~1-2Myr age of our Ophiuchus members ~2/3 of close binary systems have lost their disks, compared to only ~30% of single stars and wide binaries. This has a significant impact on the formation of giant planets, which are expected to require much longer than 1 Myr to form via core accretion and thus planets formed via this pathway should be rare in close binary systems.