A mean-field free-energy functional for an n-component mixture with an integral non-local interaction is introduced and then written explicitly for a binary mixture. We use this functional to calculate the liquid-vapor surface tension with parameters chosen to model CO2/n-C4H10 and CO2/n-C10H22, and we examine the scaling of the surface tension as a function of the difference in density between the liquid and vapor phases as various critical points are approached. Each critical point is approached on either a constant-temperature or constant-pressure path; we investigate the path dependence of the scaling behavior. For the constant-temperature paths in the CO2/n-C4H10 mixture, we compare our calculated results with experimental data. We find no significant dependence of the scaling on the path to the critical point. We note that the asymptotic scaling holds for a larger range of densities the higher the temperature of the critical point.