Can picking your nose make nostrils bigger? If not, why is one of my nostrils much bigger than the other?

There is no evidence to suggest that nose picking can affect nostril size, as it's only a temporary opening of the nose, and not something that molds it over time. There are some surgical techniques that we can use to narrow the nostril or decrease alar flaring, and you would need to consult a plastic surgeon for further discussion and evaluation. Good luck!
-David Mattos, Harvard trained plastic surgeon

Nostril asymmetries can either be a congenital condition or it can be caused by a septum deviation, but I believe that picking your nose isn't something that can affect the width of your nostrils. If there is no constant exessive and forcible pulling and streching of the nostrils, the nostrils should remain unaffected. Your problem CAN be resolved surgically, so I would definitely suggest you consulta a specialised surgeon!

Picking your nose will not make a permanent change in your nose. The assymmetry of the nostrils is caused by the inner structure of the nose. You can see a rhinoplasty surgeon and discuss your options if you would like to have it corrected. You can benefit from a rhinoplasty more than you expect as the assymmetry of the nostrils can be caused by a deviated septum which affect breathing.
Hope this helps,Ali Rıza Öreroğlu, MD FEBOPRASEuropean and Turkish Board Certified Aesthetic, Plastic and Reconstructive SurgeonIstanbul, Turkey