Jake using a napkin
A napkin was a small piece of fabric used to wipe one's face and hands clean while eating.
Napkins were in use in the mess hall and the captain's mess aboard the NX-class Enterprise NX-01. (ENT: "Terra Nova", "The Andorian Incident", "Breaking the Ice", "Fortunate Son")
In 2255 of the alternate reality, James T. Kirk used bits of wadded-up napkins to stanch a nosebleed following a fistfight with Starfleet Academy cadets in the Shipyard Bar. (Star Trek) Later, after Kirk lost command of the USS Enterprise to Christopher Pike, Pike recalled the incident with Kirk, reminding him, "You had napkins hanging out of your nose". (Star Trek Into Darkness)
In 2368, while eating at a bar on Qualor II, the Ferengi Omag shouted for a waiter. When Commander William T. Riker asked him if there was a problem, he replied that he needed more napkins. Riker told him to use his sleeve or to use the sleeve of one of his companions. Omag asked him just who he was. Riker asked him about the disappearance of a Vulcan ship and insisted he was involved, but he claimed he only dealt in warships and again asked someone to get him a napkin. (TNG: "Unification II")
Upon his release from an Argrathi prison in 2372, Miles O'Brien irrationally continued his habit of hiding food in a napkin to save it for later. (DS9: "Hard Time")
In 2373, Jake Sisko used a napkin to vomit into after his father revealed that the sauce on their food was purée of tube grubs. (DS9: "Blaze of Glory")