Can you catch a cold by sleeping with a fan on?
There are a lot of myths and old beliefs from before science fully understood the mechanism of infection by rhinoviruses (the viruses causing the common cold). The idea of being out in the cold weather or in air drafts causing a "weakening" the immune system allowing for infections isn't true.
Dowling HF, Jackson GG, Spiesman IG, Inouye T (1958). "Transmission of the common cold to volunteers under controlled conditions. III. The effect of chilling of the subjects upon susceptibility". [American Journal of Hygiene 68 (1): 59-65.];
Douglas, R.G.Jr, K.M. Lindgren, and R.B. Couch (1968). "Exposure to cold environment and rhinovirus common cold. Failure to demonstrate effect". [New England Journal of Medicine 279.]
Your immune system doesn't need to work hard to keep you warm because immune response isn't responsible for temperature regulation, so that does not put any strain on the immune system or interfere with its ability to fight infections.
The best way to avoid colds is to wash your hands regularly, and steer clear of sick people as best you can.