Nonequilibrium stimulation of superfluidity in trapped Fermi gases is discussed by analogy to the work of Eliashberg on the microwave enhancement of superconductivity. Optical excitation of the fermions balanced by heat loss due to thermal contact with a boson bath and/or evaporative cooling enables stationary nonequilibrium states to exist. Such a state manifests as a shift of the quasiparticle spectrum to higher energies and this effectively raises the pairing transition temperature. As an illustration, we calculate the effective enhancement of Cooper pairing and superfluidity in both the normal and superfluid phases for a mixture of R 87 b and L6 i in the limit of small departure from equilibrium. It is argued that in experiment the desirable effect is not limited to such small perturbations and the effective enhancement of the pairing temperature may be quite large.