Why are nosebleeds easier to get in the winter?
The vast majority of bloody noses in healthy individuals arise from one specific area in the nose (on either side): Kiesselbach's plexus.



In the mucosa of the nasal septum (the cartilagenous structure separating the nose into two sides), there is an area where several arteries "meet", giving it an exceptionally rich vascular supply, called Kiesselbach's plexus. It is on the anterior surface, and exposed to dry air and trauma. 

Bleeding typically occurs when the mucosa erodes for any reason, and the capillaries and venules (and sometimes arterioles) become exposed and subsequently break. The result is the familiar bloody dripping of a nosebleed.

Have you noticed that your lips chap more in the winter? Dry air and cols temperatures do that. The mucosa of your nose also dries out and suffers some damage. Not like your lips, but damage none the less. Damaged nasal mucosa is more susceptible to tears/trauma/erosion. If it happens in an area of rich blood supply, you get a nosebleed. 

Anything that helps with the integrity of the nasal mucosa is going to help reduce nosebleeds. One can 


run a humidifier in dry weather
avoid irritating by touching the nose or blowing nose too often
apply a very light layer of petrolatum or Bacitracin ointment to the inside of the nose covering Kiesselbach's plexus at night so the nose-breathing doesn't dry out your nose
use nasal saline spray to moisten and soothe the nasal membranes (don't rub the tip of the spray bottle against the nasal membranes, though) 


*more serious or recurrent bleeding needs medical attention to rule out  other conditions that are associated with frequent nose bleeds.

Epistaxis