We present the results of band 3, 6, and 7 ALMA observations obtained of the brightest cluster galaxy (BCG) in MACS 1931.8-2635 over the course of Cycles 4 and 5. MACS 1931 is a massive (M500 = ˜5 x 1014 M⊙) cool-core galaxy cluster in the Cluster Lensing and Supernova survey with Hubble (CLASH) with the most actively star-forming BCG in the sample, at ˜250 M⊙ yr-1. Our observations of the CO (1-0), (3-2) and (4-3) lines reveal a reservoir of molecular gas with a mass of about 2 x 1010 M⊙, on par with the Phoenix cluster, and with a complex dynamical structure. Continuum emission bands 6 and 7 also traces the morphology of cold dust in the BCG. Both the molecular gas and dust morphologies trace ultraviolet knots in the BCG core, along with an Hα-bright tail extending several tens of kpc away to the northwest. We investigate the multiphase nature of material in this system by comparing our observations of CO and dust with measurements of ultraviolet and Hα fluxes previously obtained with CLASH photometry, and discuss the correlations between molecular gas and dust morphologies, and features observed in both radio and X-ray data. We examine possible formation scenarios for dust in the BCG, as well as the implications of the velocity structure, CO line ratios, and limits on the molecular gas velocity dispersion for the mechanism of condensation of the multiphase gas in this system.