The gas permeabilities of thin liquid films stabilized by poly(N-isopropylacrylamide) (PNIPAM) and PNIPAM-SDS (sodium dodecyl sulfate) mixtures are studied using the "diminishing bubble" method. The method consists of forming a microbubble on the surface of the polymer solution and measuring the shrinking rates of the bubble and the bubble film as the gas diffuses from the interior to the exterior of the bubble. PNIPAM-stabilized films exhibit variable thicknesses and homogeneities. Interestingly, despite these variable features, the gas permeability of the film is determined principally by the structure of the adsorbed polymer layer that provides an efficient gas barrier with a value of gas permeability coefficient that is comparable to that of an SDS Newton black film. In the presence of SDS, both the film homogeneity and the gas permeability coefficient increase. These changes are related to interactions of PNIPAM with SDS in the solution and at the interface, where coadsorption of the two species forms mixed layers that are stable but that are more porous to gas transfer. The mixed PNIPAM-SDS layers, studied previously for a single water-air interface by neutron reflectivity, are further characterized here in a vertical free-draining film using X-ray reflectivity.