The verb bate has the sense of ‘decrease, lessen’, and in the phrase ‘bate one’s breath’ it describes holding back or restraining one’s breathing. The idea behind the phrase is that people in a state of extreme excitement appear to hold their breath, or even to have stopped breathing. Bate is a reduced, or ‘aphetic’, form of the more common verb abate, derived from the French verb abattre ‘to knock down’; if you think of a storm abating, or becoming still, then you will have less trouble remembering how to spell ‘bated breath’. This word is also the root of abattoir, a place where animals are 'knocked-down' – a rather genteel replacement of the earlier, and somewhat more graphic term, slaughterhouse.
The first recorded user of the phrase is Shakespeare, in The Merchant of Venice, where Shylock says to Antonio: “Shall I bend low and, in a bondman’s key, / With bated breath and whisp’ring humbleness, / Say this ”. Because bated is chiefly used today in this single expression, there’s a tendency for speakers to associate it with the more common word baited, even though there is no obvious semantic link. The verb bait is related to bite, and refers to anglers baiting a hook to catch a fish, as well as deliberately harassing or persecuting someone. To be ‘baited’, therefore, is to be set with a bait, or to be tormented. As such the phrase baited breath makes little sense. This doesn’t stop people using it, of course. There are 275 instances of baited breath in the Oxford Corpus, compared to 598 occurrences of the correct spelling bated breath. According to the Corpus, the misspelling is particularly rife on unedited blogs and in newspapers, although it is also found in published fiction. It even turns up in Harry Potter and the Prisoner of Azkaban: ‘Glaring suspiciously at Ron, Professor McGonagall pushed the portrait back open and went outside. The whole common room listened with baited breath’.