We consider the X-ray luminosity difference between neutron star and black hole soft X-ray transients (NS and BH SXTs) in quiescence. The current observational data suggest that BH SXTs are significantly fainter than NS SXTs. The luminosities of quiescent BH SXTs are consistent with the predictions of binary-evolution models for the mass transfer rate if (1) accretion occurs via an advection-dominated accretion flow (ADAF) in these systems and (2) the accreting compact objects have event horizons. The luminosities of quiescent NS SXTs are not consistent with the predictions of ADAF models when combined with binary-evolution models, unless most of the mass accreted in the ADAF is prevented from reaching the neutron star surface. We consider the possibility that mass accretion is reduced in quiescent NS SXTs because of an efficient propeller and develop a model of the propeller effect that accounts for the observed luminosities. We argue that modest winds from ADAFs are consistent with the observations, while strong winds are probably not.