Unilateral constant runny nose is very much suggestive of Cerebrospinal leak.
It may be a congenital anomaly or an acquired defect of cribriform plate were there is direct communication between subarachnoid space of brain and the nasal cavity
circulating around the brain protects the brain from mechanical trauma.
It can lead to spread of serious infections from nose to brain like meningitis
, a brain abscess
etc. in case of leaks.
Another possility is a polip in nose that irritates the local mucosa and causes watery secretions in nose.
Please have a review with your physician who would be able to decide the cause based on physical examination
I hope that answered your query.