Extensive optical and near-infrared polarimetry and photometry were used to monitor the highly polarized quasar 3C 345. Strong wavelength-dependent fractional linear polarization was observed at optical wavelengths with higher fractional polarization toward the longer optical wavelengths. Mild wavelength dependence was observed in the optical polarization position angle. When the quasar approached its peak brightness in February 1983, the polarization was 36 percent; as the object dimmed, it fell to less than 10 percent. It is suggested that wavelength dependence is caused by optically thick thermal emission, possibly from an accretion disk; this thermal emission was modeled with a blackbody. Reasonable fits to the data were found for blackbody components between 13,000 and 20,000 K. Near-infrared observations also exhibit wavelength dependence which may arise from the presence of a second polarized synchronous component. 60 references.