This is an archaic rendering, which has lost favor among translators, and with good reason. First, it basically makes no sense in ordinary English. Second, it simply misrepresents how the Pali is used. Most obviously, bhava is a countable noun (e.g. bhavaṁ aṭṭhamaṁ, “an eighth bhava”), which shows that it cannot be rendered with “becoming”, but must use either “existence” or “life”.
More significant from a doctrinal point of view, bhavataṇhā is not the desire to continually be stuck in a process of “becoming”, of changing and becoming other. It is the desire to be reborn in a permanent state of eternal bliss.
Bhava typically refers to what we normally simply call “life”, in the sense of “future lives and past lives”. Punabhava should be rendered to make this meaning explicit, as it is frequently misrepresented by interpreters who simply don’t realize that this is what it means, in contexts such as the second noble truth. Yāyaṁ taṇhā ponobbhavikā means “the craving that leads to further lives” or “the craving that leads to future existence”, or something like that. Whatever the rendering that is chosen, this meaning must be clear and explicit, as it surely is in the Pali.