As I've always interpreted it, red sunlight doesn't actively weaken Kryptonians; it's just that it lacks the frequencies of yellow sunlight that supercharge their cells. After all, Kryptonians evolved under a red sun and lived perfectly normal lives there, so it's not like it has a toxic or subtractive effect, just a neutral one. So a positive effect (yellow sun) plus a neutral effect (red sun) is basically 1 + 0 = 1. Therefore, it should be no different from being in a yellow-star system.
Now, granted, some versions show Superman only weakening gradually under red sunlight as his solar supercharge wears off (e.g. Superman: The Animated Series: "Solar Power"), while others show him or Supergirl losing their power instantly when exposed to red light (Supergirl does this). In those latter versions, we could suppose that the "natural" Kryptonian-style sunlight tricks their metabolism into reverting to normal mode or something, so their powers are immediately shut down.
But that still wouldn't matter much in a binary system. Most red stars are dwarfs, not giants, and are far smaller and dimmer than most yellow stars. Certainly a red dwarf is far more likely in an inhabited system, since the swelling of a main-sequence star into a red giant would likely have rendered any habitable-zone planets unlivable or destroyed them. So if a yellow star had a red-dwarf companion, or a distant enough red giant companion that the planet was able to survive, then the red star's light would be pretty much swamped by the yellow star's light anyway.
The only way it would be a concern is if it's a wide binary with the planet orbiting the red star, with the yellow star distant enough to be just a bright point source in the sky. That would be tantamount to being in a red-star system, with maybe a slight power boost while the yellow star was visible. Conversely, if the planet orbits the yellow star, then maybe if the red star is in the sky after the yellow star sets, and if it's close enough to be a significant illumination source and we're going with the "instant metabolic reset" theory, then it could leave Superman (or -girl) temporarily weakened while it was in the sky.