The editors for the 1920 LDS edition apparently thought that the adverbial up (which is extant in the original manuscript) was inappropriate here, so it was deleted from the 1920 edition. The change was intentional since it was marked in the committee copy.
This editing is consistent with usage elsewhere in the text referring to drowning; that is, the adverbial up does not occur with the expected in preposition:
Nonetheless, one could argue for the up here in Alma 63:8 since one can think of the ocean waves rising up to cover the ship and its passengers.
There is evidence for the expression in American dialectal speech, as in the following example from the late 19th century:
Literature Online lists a few examples of drowned up dating from the 1600s (here I regularize the spelling and ignore all other accidentals):
For each of these 17th-century examples, the meaning seems to be a metaphorical ‘covered up’ (usually with water). On the other hand, the instance of drowned up in Alma 63:8 literally refers to death by drowning (as do all the other instances of the verb drown in the Book of Mormon). Despite these differences, it does appear that the up is intended in Alma 63:8. Moreover, there just isn’t that much of a problem having the up in this sentence. The critical text will restore the up.
Summary: In accord with the reading of the original manuscript, restore the up in Alma 63:8: “and we suppose that they are drowned up in the depths of the sea”; this use of up with drown is unique for the text but appears to be fully intended.