There are a good many CGs in the Mammoth Cave area. If you don't want to stay in the park itself, there are two nearby that I would personally recommend.
Singing Hills Campground has really been fixed up nicely in the last couple of years. It had been fairly run down, but the present owners have done a fine job of fixing it up and did it right. They have Wi-Fi, but no cable. However, if you have a decent antenna, you should get plenty of TV stations if you want them. Here's their website: http://www.singinghillsrvpark.com/
The other -- and less scenic but still nice -- is Cave Country RV Campground. It's right off of I-65, so there is some noise, but not intolerable. Being close to the interstate, you'll have easy access to restaurants and Bowling Green is just a few miles south down the road if you want to go shopping or something. Wi-Fi and Cable are available. Here's their website: http://www.cavecountryrv.com/
Also close by (in Bowling Green) is the National Corvette Museum and the GM assembly plant where Corvettes are built. Not sure if they still do plant tours, but the museum is a lot of fun. I try to visit whenever I'm down that way.
Hope this helps!