Both terms are correct, but runny nose is much more common.
Runny nose is the idiom that Americans use to describe what happens when your nose runs (that is, when liquid comes out of your nose because of a cold, allergy, or crying). This is the term for the general condition of having a running nose and is used in a sentence like the word cold when it refers to an illness.
Note also that the idiom is to have a runny nose:
My daughter has cough and a runny nose.
Symptoms include runny nose, itchy eyes, and scratchy throat.
I wish I could do something about my runny nose.
Running nose has the same meaning but is used only to refer to a specific instance of this symptom, in contrast to the condition in general. This expression is used very rarely; it's literal and descriptive and can be understood as being too direct a reference to something that is unpleasant:
My uncle wiped his running nose with his sleeve.
Her baby has a perpetually running nose and a rash on his stomach.