We analyze the interplay between vibrational and electronic degrees of freedom in charge transport across a molecular single-electron transistor. We focus on the wide class of molecules which possess quasidegenerate vibrational eigenstates, while no degeneracy occurs for their anionic configuration. We show that the combined effect of a thermal environment and coupling to leads, involving tunneling events charging and discharging the molecule, leads to a dynamical symmetry breaking where quasidegenerate eigenstates acquire different occupations. This imbalance gives rise to a characteristic asymmetry of the current versus an applied gate voltage.