Before we even get going, consider the description of Billy being hanged, that is to say: drawn up by the neck until he is dead. The narrator says,
The hull, deliberately recovering from the periodic roll to leeward, was just regaining an even keel when the last signal, a preconcerted dumb one, was given. At the same moment it chanced that the vapory fleece hanging low in the East was shot through with a soft glory as of the fleece of the Lamb of God seen in mystical vision, and simultaneously therewith, watched by the wedged mass of upturned faces, Billy ascended; and, ascending, took the full rose of dawn.
In the pinioned figure arrived at the yard-end, to the wonder of all no motion was apparent, none save that created by the slow roll of the hull in moderate weather, so majestic in a great ship ponderously cannoned. (25.6)
Ahem. Ahem. WHAT?!? We don't know what you think, but to us this makes Billy sound like an angelic little kid jumping on a trampoline on a hot summer day, his face taking on "the full rose of dawn." Let us re-state the facts: Billy was falsely accused by John Claggart. He was unable to respond to the accusation because of his stutter. Out of frustration, he struck Claggart, accidentally killing him. Although Captain Vere sympathized with Billy and saw that it was unjust, he considers it his duty to condemn Billy to death. Billy is, again, being hanged.
This passage sounds like it was written by Captain Vere many years later, trying to seek solace in religious imagery. It, quite frankly, romanticizes Billy's execution. We acknowledge that we're stating the case pretty harshly, but we're doing so just to contrast the biases and assumptions that get packed into the narrator's style. Maybe Billy was a remarkable man and could understand Vere's dilemma, and maybe he wasn't even afraid of death. But all the same, can you imagine him describing his own execution this way?
Consider the narrator's word choices. He describes the horizon as "the vapory fleece hanging low in the East." Why does he choose the word "fleece?" Because he is trying to prepare the reader for a comparison between Billy and Jesus Christ, who is seen as a sacrificial lamb in Christian theology. A moment later, the narrator goes on, "as of the fleece of the Lamb of God seen in mystical vision." The narrator's tone, his desire to portray Billy in a religious light, manifests itself in his various word choices.
Also consider the back-and-forth nature of the sentences. They swing from one clause to another like a ship getting tossed in a storm, but they can't even say exactly what they mean. The narrator describes Billy's hanging as "Billy ascended; and, ascending, took the full rose of dawn." The sentence has a nice lilt to it, a nice rhythm, but all he's saying is that Billy is hanged. It's almost as if he can obscure the violence of what happens by making his sentences flow with a poetic sense and choosing words that have as much symbolic significance as descriptive value.
We've made our bias clear. We think the narrator's moralistic prejudice comes through most clearly in his very style, a style far removed from the language that Billy Budd himself would have known and spoken. But note that this is not a criticism of Melville. We have little doubt that Melville was aware of the over-wrought moralistic portrayal of events, and what we're doing is criticizing the fictional narrator of the story. We don't think of the narrator as Melville himself, but as Melville's fictional creation. You can check out "Narrator Point of View" for more specifics on the story's narrator.
Lastly, we'll note that the narrator closes the story with a vulgar sea poem. To a large degree, he seems to be poking fun at its sentimental style, at the poor writing. But let us ask: is his style any better?