Worf babysitting Kirayoshi
Babysitting was the act of minding children occasionally while their parents were otherwise occupied. One who babysat was known as a babysitter.
In 2153, Travis Mayweather remarked that you couldn't go ten meters on a cargo ship without running into your babysitter or a great-aunt. (ENT: "Horizon")
In 2365, Brenna Odell considered herself an authority on babysitting, and therefore security, assuming the two were similar. (TNG: "Up The Long Ladder")
In 2373, Miles O'Brien was left without a babysitter after Annel canceled on him. Jake Sisko offered, saying he'd watched Ensign Vilix'pran's hatchlings before. When Kirayoshi cried in Jake's arms, O'Brien took him back, which shushed the baby. This was repeated later in Quark's during a darts game with Julian Bashir, who expressed surprise that O'Brien couldn't find a replacement sitter. Eventually, Worf was left to mind him while O'Brien got some much needed rest. (DS9: "Business as Usual")
Also that year, Q responded to Kathryn Janeway's honor at being asked to be his baby's godmother by saying "wait until we ask you to baby-sit". (VOY: "The Q and the Grey")
In 2374, Jadzia Dax offered to watch Kirayoshi while the O'Briens spent time with a time displaced Molly. Worf then took over so Jadzia could run a spectral scan on a comet going through the Denorios belt. He introduced the boy to the Hehh-duHpp, something which left an impression on the child. Later, when Jadzia expressed concerns that he was taking it too seriously, he told her that he was trying to prove himself capable of being a parent in her eyes. (DS9: "Time's Orphan")
In 2377, Tom Paris assured a pregnant B'Elanna Torres that Neelix was more than ready to babysit their child when it was born. (VOY: "Lineage")