The Board of Trade (BoT) records (Public Record Office groups BT158, BT159, BT160) which cover births, deaths and a few marriages on British registered ships from 1854 to 1890 are available from the LDS on four films 1419469, 1419470, 1419471, 1419472. These are not well organised, or usefully indexed and an event may be recorded on any one of the four reels.
However these are real registers (sometimes with gossippy comments) which make for interesting reading.

Later BoT BMD records in group BT334 (1890 to 1972) don't appear to have been filmed by the LDS.

These Board of Trade records were supposed to be passed to the Registrar to be used in a GRO (General Register Office) series. Sometimes this was not done, and sometimes entries were made directly to the GRO so are absent from the BoT registers. Also the GRO series started earlier.

GRO Births at sea from 1837 to 1965 are available on 56 microfiche; however these are only indexes which contain names, dates and in some years, ship names. LDS Fiche numbers 6137268 - 6137322 The LDS Family Search Library Catalog will identify which fiche relates to which year.

GRO Deaths at sea from 1837 to 1965 are on 160 microfiche, again these are only indexes. LDS Fiche numbers 6137189 to 6137266. The LDS Family Search Library Catalog will identify which fiche or fiches relate to which year.
Observant readers may wonder what happened to microfiche 6137267. This is listed separately as:
Index to informal certificate of births at sea (1-41) 1839 - 1867

Marriages at sea were really rather rare and the GRO doesn't have a separate series. There is an RG34 series of miscellaneous foreign marriages for British nationals and at the end of the places named, the list adds - and at sea. I've no idea how many 'at sea' marriages are on the film. LDS have filmed these:

volume 1, 1826-1870 film number 1818071
volume 2, 1871-1895 1818072
volume 3, 1896-1900 1818073
volume 4, 1900-1905 1818074

Apparently BoT were passing marriage entries to the GRO but GRO were not processing these and were actually denying they had them! (source: mariners list)

The LDS catalog lists many other record groups which may contain a few BMD at sea events but the above seem the most likely. Note that these records generally refer to events on British registered ships whether the people are British or not. Also the ships need not have been trading between British (Empire) ports; so a birth on a Cunarder sailing from Trieste to New York may be recorded.

I'd like to thank Debbie Beavis of the Mariners List for supplying the information about record groups but the connection to LDS numbers is my work, so please blame me if the film or fiche numbers are wrong.

The Registrar-General of Births, Deaths, and Marriages in England, hereby directs the attention of all Captains or Commanding Officers of British Vessels to the following Provisions of the Act of the 6th and 7th of William IV. cap. 86, entitled " An Act for Registering Births, Deaths, and Marriages in England. " namely :—
In the 21st Section it is enacted, " That if any Child of an English Parent shall be born at Sea, on board a British Vessel, the Captain or Commanding Officer of the Vessel on board of which the said Child shall have been born shall forthwith make a Minute of the several Particulars herein-before required to be inserted in the Register touching the Birth of such Child, so far as the same may be known, and the name of the Vessel wherein the Birth took place, and shall, on the arrival of the Vessel in any Port of the United Kingdom, or by any other sooner opportunity, send a Certificate of the said Minute, through the Post-Office, to the Registrar-General. "
The following are the Schedules containing the particulars required to be inserted in the Register, so far as the same may be known :—

The word " English " includes only Natives of England and Wales. The word " British " includes Vessels belonging to all parts of the British Empire.
The Minute may be made in the Ships's Log, and the Certificate of the Minute shall be a true copy of it, with the following words, or others to the like effect, subscribed :—
  " I hereby Certify that this is a true copy of the Minute made in the Log of          on the     day of       18   .

Such Certificate to be sent through the Post-Office, directed—