We present 800 mu m continuum photometry of pre-main-sequence binary stars with projected separations a(p) < 150 AU in the Scorpius-Ophiuchus star-forming region. Combining our observations with published 1300 mu m continuum photometry from Andre & Montmerle (1994), we find that binaries in Sco-Oph with 1 < a(p) < 50-100 AU have lower submillimeter continuum fluxes than wider binaries or single stars, as previously found for Taurus-Auriga binaries. The wide binaries and single stars have indistinguishable submillimeter flux distributions. When the Sco-Oph and Tau-Aur samples are combined, this dependence of submillimeter flux on binary separation is detected with a confidence level of greater than 99%. Thus, binary companions with separations less than 50-100 AU significantly influence the nature of associated disks. We have explored the hypothesis that the reduction in submillimeter flux is the result of gaps cleared in 100 AU disks by companions. Gap clearing produces the qualitative dependence of submillimeter flux on binary separation, and a simple model suggests that large gaps in disks with surface densities typical of wide-binary or single-star disks can reduce submillimeter fluxes to levels consistent with the observed limits. This model shows that the present submillimeter flux upper limits do not necessarily imply a large reduction in disk surface densities. Two-thirds of the pre-main-sequence binaries were detected by IRAS at 60 mu m, showing that most binaries have at least one circumstellar disk. We have used these fluxes to place lower limits of 10(-5) M. on circumstellar disk masses. Similarly, the 60 mu m fluxes indicate that the circumstellar disk surface densities are no more than 2 orders of magnitude smaller than those of typical disks around single stars. Our upper limits on submillimeter fluxes place upper limits of 0.005 M. on circumbinary disk masses. Thus massive circumbinary disks (such as that found around GG Tau) are rare among binaries with projected separations between a few AU and 100 AU. Circumbinary disks are found around some binaries with separations less than a few AU.