Molecular strong coupling offers exciting prospects in physics, chemistry, and materials science. While attention has been focused on developing realistic models for the molecular systems, the important role played by the entire photonic mode structure of the optical cavities has been less explored. We show that the effectiveness of molecular strong coupling may be critically dependent on cavity finesse. Specifically we only see emission associated with a dispersive lower polariton for cavities with sufficient finesse. By developing an analytical model of cavity photoluminescence in a multimode structure we clarify the role of finite-finesse in polariton formation and show that lowering the finesse reduces the extent of the mixing of light and matter in polariton states. We suggest that the detailed nature of the photonic modes supported by a cavity will be as important in developing a coherent framework for molecular strong coupling as the inclusion of realistic molecular models.