I agree with dgruss23. If Jupiter had enough mass to form a red dwarf, it would still be a red dwarf. To rephrase what he said, since they are so tiny compared to normal stars, they fuse hydrogen at a much slower rate. Red dwarf stars have a very, very long lifespan. Also, I'll have to check my book, but even a tiny red dwarf is upwards of 80 Jupiter masses.

Based on that, my opinion is that Jupiter is not a red dwarf.