It may depend on what country the ship was coming from. For example: Here is an excerp from Royal Museums Greenwich at: http://www.rmg.co.uk/researchers/library/research-guides/general-introduction/research-guide-a3-tracing-family-history-from-maritime-records
Births and deaths at sea
Details of births and death at sea aboard British-registered ships were, since the introduction of civil registration (1 July 1837 in England and Wales; 1 January 1855 in Scotland; and 1 January 1864 in Ireland), were required to be sent to the appropriate General Register Office depending on the nationality or normal place of residence of the father or of the deceased. (Note: records related to foreigners were, from 1875, sent to the GRO for England and Wales.)
Until 1874 these were sent directly by the ship’s captain; after that date they were transmitted via the Registrar-General of Shipping and Seamen. Each General Register Office maintains a Register of Marine Deaths with a separate index: most of these indexes are now online through several commercial organisations.
The primary record of a birth or death at sea is the ship’s official documentation. This usually means the ship’s official log, which was introduced in October 1851, but until the early 1890s may be found on the back of crew lists, including those of passengers.
Information on births and deaths at sea recorded in log books was extracted by the Registrar-General of Shipping and Seamen and entered in registers now to be found at The National Archives, in series BT 153–BT 160 covering the period 1854–1890 with a few later entries. These are being digitised and are becoming available: details are to be found on The National Archives website.
From 1891 details were copied into consolidated indexed registers kept by the Registrar-General of Shipping and Seamen: series BT 334 at The National Archives. These should cover both births and deaths that actually occurred at sea as well as on-board in port or in a local foreign hospital. Those where the event actually occurred at sea would have been reported to the General Register Office for England and Wales. Where the deceased or father was Scottish or Irish, a copy was sent to the General Register Offices in Edinburgh or Dublin.
A record of any births or deaths on a ship inbound to the UK, originating from a port outside Europe or the Mediterranean, had to be recorded on a passenger list. Those that survive (1890–1960, with a very few earlier) are in series BT 26 at The National Archives and are available online at ancestry.co.uk."
Personally, I would document everything that I know and keep looking for more details over time.