Not on the Fort Jackson portion for sure. I'm stationed here and there is 100% no metal detecting allowed within the limits of Fort Jackson. The rest of it is public and I don't think that's a big issue.
I did not know if they were a part of the state park system. You can't metal detect in the state parks so if they are a part of the parks then it would not be allowed. I can't tell by looking at their web page if they are or not.
I don't think they are either. If it doesn't say then it might not be an issue. If it is then you can say you researched it and couldn't find an answer.