Other than Prufrock's avoidance of the question, what else shows him to be a coward?
There is the epigram which is from Dante’s Inferno. It is Dante meeting with Guido da Montefeltro where Dante is requesting advice form Guido. Because Dante is speaking to someone in the underworld and no one has ever returned from the underworld, Dante knows this conversation will never be repeated. One could interpret that Elliot used this epigram to show how Prufrock never intended this monologue to be repeated. It is ambiguous who Prufrock is speaking to: the reader? A friend? The woman who he is asking or attempts to ask to marry him? It could also be an internal monologue, a poetic stream of consciousness. If that’s the case, then Prufrock lacks confidence so much that he is even tentative in his own mind; where no one but he can hear himself.
What else shows him to be a coward, or to be more sympathetic, overly analytical and lacking confidence; some critics have even suggested a multiple-personality disorder?
The epigram refers to Dante and the underworld. Later in the poem Prufrock refers to himself as Lazarus, as if this quest to ask her this question is as grandiose and miraculous (for him) as coming back from the dead. So, besides the fact that he never asks the question, there are the psychological lengths he goes to in attempting to psyche himself into asking her.
Going back to the epigram, if he is telling this to a friend, the reader or internally and never intending to repeat it, then he never truly intends to ask her the question. The last stanza can be interpreted in different ways (so can each stanza, really). But in this analysis, note that Prufrock starts using ‘we’ instead of ‘I.’ Maybe he has asked her and it worked. But since going through different conceptions of himself, the we may refer to the ‘we’s’ of himself and the mermaids are in his mind; only to be awakened by real human voices, which is when his internal conflict and emotions will drown, or that he will drown in the real world as opposed to his internal dialogues with himself. In any case, unless the ‘we’ means he succeeded, he has never posed the question.