It depends on the state. Here in MA, if a company or anybody does a thorough background check, the arrest may pop up. My question to you is, did you take a plea and put on probation and the case dismissed after the probationary period? If that is the case, it may show up.
If charged with a DUI, you should have been arrested. If not arrested, but cited and then the charges were dismissed, the charge could likely appear on your record. If the charges were dismissed, then the charge (and arrest if there was one) can be expunged by filing a motion with the original court. It is a simple process and once expunged the charge will be erased from your record.
Good luck, and feel free to contact me if you would like to discuss your options as I am currently working on filing a few related motions.
If the case was formally filed and then dismissed, then it is on your background record and will remain there unless you take the appropriate action. Even if it was not filed, then it remains on law enforcement records and is likely to show up whenever you are stopped by the police, [significantly increasing the odds that you will be arrested again]. However, if it was never filed in court, it will not show up in the typical background check, but will appear in some background checks that go farther than others. In any event, after 2 years from the arrest-and if you were not arrested for anything else at the same time, you may seek an expunction of the arrest, which will remove it from ALL records, which is what you want. There are provisions for removal before 2 years but they are so difficult that it is generally recommended that you wait the 2 years and then have an attorney get you an expunction.