The expression with bated breath is recorded as appearing for the first time in William Shakespeare’s The Merchant of Venice towards the end of the 16th century. It refers to having subdued or restrained breathing because of some strong emotion.
The verb to bate, which means reduce the intensity of; take away; or lower in estimation or amount (an archaic usage), has long since disappeared from common use.
So it’s no surprise that it’s often misspelled as baited. According to Merriam Webster, bait means to persecute, harass or lure.
Here’s an example to illustrate the two meanings: She waited with bated breath to see if he would take the bait.
Here are some quotations illustrating their usage:
… the end of the year approaches, practically everyone in the fashion, beauty and interior design industries waits with bated breath to see what Pantone will proclaim as the official color for the upcoming year. (www.chicagotribune.com)
… the world — or at least, the American public — waited with bated breath on Thursday for the National Archives to release its final batch of records related to the assassination of … (www.usatoday.com)
… Channel and blue catfish are good on hot dogs, shrimp, Spam, and frozen shad. Yellow catfish are good on juglines baited with live perch. (www.chicagotribune.com)