There are plenty of companies that keep, and hire pilots who have DUIs. I have a friend who was hired by USAirways even though she had a DUI and could not do Canada trips. Many airlines have a HIMS program for pilots who have had issues with alcohol, including DUIs.

On a somewhat related note, I have a Canadian friend who recently became a naturalized U.S. citizen because he had heard of a few stories of U.S. Green Card holders being stopped at the border, being denied entry, and ultimately having their Green Cards revoked. While his naturalization was being processed, he had to inform the company that he was unable to do international trips, and there were no issues. I say it is somewhat related to the HIMS issue because he was instrumental in setting up the HIMS program at my old company. I also know some people on the European side who have had some issues getting their U.S. C1/D visa issued or renewed, and the company adjusted their schedules to avoid U.S. trips during that period.

So based on what I have seen, airlines are used to and understand the occasional passport/visa/travel restriction issues and in most cases can work around them for at least a period of time.