My husband's grandfather was born at sea in 1899. His mom was pregnant when she boarded the boat and no longer pregnant when she left it. Not all that uncommon.
From what I can tell, he was considered an American citizen from birth, even though he was not in American waters (in fact,the boat went to Canada...the family's destination was the US though; they just took a boat to Canada first).
I have not found his birth certificate but all his subsequent paperwork says he was born in the US (in DC where the family lived) and he seemed not to need to naturalize. I have the ship's records that document both the lack of said grandfather when the family embarked and the notations on the pre-written record of his date of birth and given name.
I would look for the immigration records first (the ship passage). If it's before 1905 or so, you should get good records. Earlier ones are spotty.