The sense of ‘still’ that works in this line is:
still, adv. 3.a. With reference to action or condition: Without change, interruption, or cessation; continually, constantly; on every occasion, invariably; always. Obsolete exc. poetic.
Oxford English Dictionory
So Byron means that the monsoon wind quietly blows “her steady breath” without interruption for some months. But in context, the choice of phrasing is unfortunate, because you can’t use ‘still’ when describing the weather without calling to mind a more common sense of the word:
still, adj. 1. Motionless; not moving from one place, stationary; also, remaining in the same position or attitude, quiescent.
which is quite the opposite of what is meant. The impression I get is that the difficulty of writing ottava rima in English, where there are not nearly so many rhymes as in Italian where the verse form originated, causes Byron to get himself into syntactic and semantic tangles, and sometimes he solves them by throwing down his pen and leaving the mess for the reader to figure out. In this stanza, however, the rhymes are so inventive that maybe we don’t mind that the sense has gone astray.
The word ‘still’ is italicized to tell the reader where to put the stress so that ‘still is’ rhymes with ‘Achilles’ and ‘kill his’. (As pointed out in a comment by Peter Shor.)