Family Search has added the UK Maritime Births, Marriages & Deaths 1787–1933 collection to its huge database. The details returned from a search are limited to the person's name and the date of the event, together with the name of ship.
If you want to investigate further, you are referred to FindMyPast to view the full details and an image of the original record.
Obviously, given the start date, this collection includes many bmds of Irish people, but it consists of only 42,183 entries in total so it isn't going to solve too many mysteries. However, it's always worth checking, even if the standard of recording wasn't exactly fulsome.
Birth information is pretty scant (example: a child was born to the wife of Sergeant Driscoll on 4 November 1859 on the Ealing Grove), but there's a bit of an improvement with marriages (example: Miss I Doherty, a stewardess, married Paul Striebeck, a single, 32-year-old seaman, on board the ship Nilverton on 11 June 1899, the initials of the bride's father being H and the groom's father F).
Deaths, don't typically provide much genealogical data (a significant proportion are of infants and young children, in any case) but you might find an occasional death worth following up if it seems to connect with a family tale (example: Daniel Driscoll, aged 40, drowned on 26 August 1878 while on the ship City of Brussels).
The collection can be found here on Family Search or you can search at Find My Past.