Whose nose doesn't drip when you're out in the cold? This common situation occurs when cold, dry air irritates your nasal lining. Your nose's glands respond by pumping out more mucus than normal to keep it moist so the lining doesn't dry out.
But don't assume the chilly temps are the sole reason your nose is drippy. Colds and upper respiratory infections are common in winter. In fact, it's a person can have multiple colds in a row. So it might be the cold, or you might have a cold -- or both.