A runny nose was the leakage of mucus from one's nostrils. One of the symptoms of Levodian flu was a runny nose. (VOY: "Tattoo")
In 2370, while camping in the Gamma Quadrant with Benjamin Sisko, Jake Sisko, and Nog, Quark complained of a runny nose, from which he was suffering due to an allergic reaction. (DS9: "The Jem'Hadar")
In 2371, The Doctor bemoaned runny noses, stubbed toes, and pimples on one's cheek as "the trivia of medicine". (VOY: "Parallax")
In 2372, while suffering from a holographic case of Levodian flu, The Doctor used holographic tissue paper to wipe his runny nose. (VOY: "Tattoo")