As the Wikipedia article on mucus states, cold weather does cause runny noses:
During cold weather, the cilia, which normally sweep mucus away from the nostrils and toward the back of the throat (see respiratory epithelium), become sluggish or completely cease functioning. This results in mucus running down the nose and dripping (a runny nose). Mucus also thickens in cold weather; when an individual comes in from the cold, the mucus thaws and begins to run before the cilia begin to work again.
Mucus also functions as a trap for bacteria and viruses (again, quoting Wikipedia):
A major function of (mucus) is to protect against infectious agents such as fungi, bacteria and viruses. ... In the respiratory system, mucus aids in the protection of the lungs by trapping foreign particles that enter it, in particular, through the nose, during normal breathing.
I don't have any source on this, but I should think that a runny nose would be no better or worse than a dry nose for protecting against viral infections. If anything, I think it would be better to have a runny nose, as bacteria and viruses would have to "swim upstream" to reach their target destination.