A dishwasher was an individual who was employed to wash dishes.
In describing the employees at the Hotel Brian in 1893, Jack London (a bellboy there), described the dishwasher there as being "drunk all day, but at least he gets here on time." (TNG: "Time's Arrow")
After having time traveled back to 1930, James T. Kirk by employed by Edith Keeler at the Twenty-First Street Mission, where he helped by doing dishes, sweeping, general cleaning, at a rate of 15 cents an hour for 10 hours a day. (TOS: "The City on the Edge of Forever")
During the 1950s, on Earth, Negros had very few job options in Harlem, according to Jimmy, suggesting delivery boy or dishwasher as being two of the few options he had to choose from. (DS9: "Far Beyond the Stars")
By the 24th century, dish washing had changed significantly, and simply involved placing the dirty dishes back into the replicator for reprocessing. (DS9: "Prophet Motive", "Hard Time", "Nor the Battle to the Strong"; VOY: "Memorial")