Employers typically run background checks on potential employees. Unfortunately, when someone is arrested for a DUI, even if their record is expunged and there is no official copy of the arrest that would show up on a traditional criminal background check, a mug shot from the arrest might be picked up by the third-party companies use for this process. It is difficult to completely clear a record given the way the internet works. Even a first-time DUI arrest can cause problems for people as a result of the new technology associated with background checks.
That being said, the only way a conviction can get an employed person in trouble at their job is if their employer were to find out about it. If a person was arrested and taken to the Charleston County Detention Center, then their mug shot would be posted online after their arrest. If a potential employer knows where to look, they could find out about that arrest, and it is up to each individual employer how they deal with the situation.