I also found a reference to the Commissioners of Emigration which revealed that they are New York State Commissioners of Emigration. Could their reports be at the State Archives as well as the NYC ones? [Perhaps not -- (At first, I thought that their reports may have been lost in the 1911 fire at the State Library of New York.])
On a prior answer, I left this comment (posted Jul 19, 2015 at 22:59):
According to USCIS historian Marian L. Smith, the only records which survive for NY arrivals before 1897 are US Customs House records, created by the US Customs Service. The Castle Garden and Barge Office passenger lists (which would have had immigration information like the later lists) were transferred to Ellis Island and were destroyed in the 1897 Ellis Island fire.
Note also that while we think of the manifests required by the US as "arrival lists", during some periods they were usually filled out at the port of embarkation and amended and annotated as necessary. They were checked (not filled out) as people disembarked. For this particular problem, the information about who was born and died on the voyage would have been added at a later time than the rest of the information on the list; the same would hold true for the later passenger lists from Ellis Island that have lists about people being detained.
For this particular question, the surviving lists are US Customs House lists. I don't know as much about the process by which these lists were created as I do about the later passenger lists, but differences in handwriting might provide clues about what information was filled out first (e.g. if the list was created in the departure port) and what was added later.
As for other company records, I would expect that if a crewman died, his name would be listed in the ship's log, but I don't know if the ship's own log would list the birth and death of passengers. My questions are: where was the William Stetson registered, and what company was in charge of the ship? The home port might determine what regulations were in effect, and what information would be required in the log. There might be correspondence in the BT records held by TNA and/or in any company archives that survive for the shipping company.
