Linguist here.

tl;dr: the exact reason is lost to history, possible reasons below.

One popular rationale is that ships are traditionally given female names, hence, you might refer to it as a she since its name is female. In ancient times, it was common to dedicate a ship to a goddess, but at the rise of Christianity, belief in the goddesses waned, and they largely replaced those dedications with dedications to their mothers or the queen.

As a linguist, I'm more inclined to believe the following, also a serious academic proposal: in most Indo-European languages, the gender of the noun ship is feminine. Unfortunately, in Old English, scip is neuter. That's not a huge problem though, as neuter seems to be the exception and feminine the norm. We can postulate that OE scip was originally feminine and "lost" its grammatical gender.

(It could be argued that when the Proto-Indo-European genders "animate" and "inanimate" were evolving into the three we know today, that the word for ship fell into different categories of "feminine" and "neuter" based on the language. Note that my use of PIE above is meant to represent the entirety of its constituent branches and not the reconstructed language itself. Anyway, grammatical gender is weird, you end up with things like "sailors" being feminine in Greek and Latin even though you only ever had male sailors. The best explanation I've read for this is that in PIE, when the inanimate gender was evolving into feminine and neuter, that the feminine gender is actually a "collective" gender, so much of livestock is given that grammatical gender. It would have been during this period, while the gender was thought of by the native speakers as "collective" rather than "feminine," that nauta would have been labeled as part of a collective, thus when the gender changes to "feminine," the nauta is grandfathered in. I'm not sure I wholly buy that theory, but there's simply too much we don't know about the proto-languages.)