The biblical theme seemed to come up a lot in Billy Budd. The story of Adam and Eve in the Garden of Eden especially.
The narrator relates Billy with Adam multiple times. One specific time was on page 52, "Billy in many respects was a little more that a sort of upright barbarian, much more perhaps as Adam presumably might have been ere the urbane Serpent wriggled himself into his company."
I think that Adam represents Billy while the Serpent represents Claggart. Billy is innocent, naive, and simple from the start of the book. Claggart, while appearing to have a good heart, turns out to actually be rather evil. The Serpent is described as urbane, meaning polite. Yet the Serpent in the Bible is often referred to evil.
What I took this as was that Claggart appears to be polite but below the surface, he in actuality is evil.
Another smaller connection to support the biblical theme in the Bible and in Billy Budd could be that Claggert communicates to Vere who in turn speaks to Billy which eventually will lead to his fall. While in the Bible it starts with the Sepent speaking to Eve, and continuing on to Adam which leads to his fall.