The original manuscript has the subject pronoun he after the conjunction but. While copying into the printer’s manuscript, Oliver Cowdery accidentally dropped the pronoun. As noted under Mosiah 24:11, either reading will work here in Alma 51:25. More specifically, we note that elsewhere in the original text, in three cases of an initial clause beginning with “it came to pass” followed later by a but-clause, the subject is explicitly repeated (as a pronoun), but in four cases the potentially repeatable subject is ellipted:
repeated subject
ellipted subject
In these examples, the subject can be explicitly repeated as a pronoun or ellipted. In the case of Alma 51:25, the critical text will therefore follow the reading in 𝓞, the earliest textual source.
Summary: Restore the subject pronoun he in Alma 51:25 (“but he kept them down by the seashore”).