Coherent upconversion of terahertz and mid-infrared signals into visible light opens new horizons for spectroscopy, imaging, and sensing but represents a challenge for conventional nonlinear optics. Here, we used a plasmonic nanocavity hosting a few hundred molecules to demonstrate optomechanical transduction of submicrowatt continuous-wave signals from the mid-infrared (32 terahertz) onto the visible domain at ambient conditions. The incoming field resonantly drives a collective molecular vibration, which imprints a coherent modulation on a visible pump laser and results in upconverted Raman sidebands with subnatural linewidth. Our dual-band nanocavity offers an estimated 13 orders of magnitude enhancement in upconversion efficiency per molecule. Our results demonstrate that molecular cavity optomechanics is a flexible paradigm for frequency conversion leveraging tailorable molecular and plasmonic properties.