Well, chronic rubbing certainly can result in chapping of the nose. And, wetness from a runny nose can also result in a chapped, red, scaly appearance. So, first things first: We've got to figure out a way to fix the root of the problem and decrease the amount of chapping by decreasing the rubbing or decreasing the runniness of your nose.
The second step is to use something mild. Something like a mild hydrocortisone can help prevent this, which you can get over the counter. A 1% hydrocortisone used for a week or two can really help with some of the inflammation secondary to chronic running and wetness of the nose.
If the hydrocortisone 1% that you use for a week or two does not work, then go see your dermatologist for something a bit stronger.