Mama Gerties is lovely and affordable. Almost nothing to do for kids though. They have private stream side sites. We also loved Asheville East KOA and they have nice sites on the Swannanoa River--narrow sites but very long. Close to the highway and sporadic train noise--but that didn't bother us. Very nice owner there and good staff. Both of these campgrounds are fairly convenient to Asheville (15 minutes?) but they are also nice because they are close to Black Mountain--which is almost like a mini-Asheville.
My in-laws live there so we have been to Asheville almost twenty times. IF you want recommendations for food or sites to see let me know.
We spent a full week at Asheville East KOA this past summer and have many pictures of it on our blog if you want to scroll down to July and take a look.
Best--Jeremy