Why do we get runny noses in the cold?
There are two reasons for this:


Nasal mucus helps warm inhaled air before it reaches the lungs. In cold weather, the mucus tends to dry out, so the membranes increase their production.
At the same time, exhaled air is warmer than the surrounding air, so it contains more moisture than the outside air can hold. This moisture condenses around the tip of the nose.


Explanation found here.

So there's no particular advantage to getting a runny nose; it's just a normal reaction occurring in extreme conditions.