Contrary to popular belief, the airlines aren't looking for all "saints". Many have busted one, two, and even three checkrides and have been able to get hired at the regionals and majors. A lucky few have even been hired with a previous DUI on their record although this if anything should be viewed as a more serious and show stopping event than that of a busted checkride. One busted ride certainly won't stop you from being able to obtain employment at the airlines, I have known many who were successful and many times the busted ride was never even discussed during the interview process. Just be able and ready to "justify" the bust, explain it, and most importantly that you learned from it and were able to move on and be successful after the experience. Don't make a habit out of it, I once flew with a first officer who busted his private, CFI, MEI, initial 135 sic ride, and upgrade ride with the fsdo, this would be no joy in trying to explain it during the interview process should this topic be brought up by the interview board. One busted ride, not a problem but multiple busted rides may raise a rather large red flag in the eyes of the interview board since they may question whether or not you can be successful in passing a 121 training program.
I wouldn't be overly concerned about it at this point, it may or may not come up during the interview process. Just make certain that you don't try to hide this since it is noted in your airman file and you likely will be let go if you try and cover it up.
Washing out of a 121 or 135 training program is more serious which you want to avoid.