From what I understand, the KOA just south of Watkins Glen, on state route 414, is a nice campground. I only stayed there once and that was in 1973, but just by driving by, it looks like it's kept up nicely and usually looks filled during the summer months.
There is a private campground south of the Watkins Glen KOA called Ferenbaugh. It's more near Corning, NY than Watkins Glen, but Watkins is only about a 20 minute drive from Ferenbaugh. I also camped there many years ago, so again I can't really make a judgement call on how it is today. But when driving by, there are always campers there. And it's only about 10 minutes from the Corning Museum Of Glass.
Clute Park Campground, as stated in an above post, is just on the outskirts of the village of Watkins Glen. It is located, just across state route 14, from the southern shore of beautiful Seneca Lake and the lake side Clute Park. It's within walking distance of the village.
There's also is a New York state campground at the top of Watkins Glen state park. I've know more than a few people that's camped there in the last couple of years and they always told me it was picturesque and they enjoyed it. A walk through the gorge there along with all the water falls is a real experience.
There's a lot to see and experience in the small village of Watkins Glen. The state park with its water falls, Seneca Lake, the Village stores and the many wineries within the Finger Lakes area are within a 30 minute drive from the village. There are also boat tours of Seneca Lake including an old wooden Schooner, "True Love." The Finger Lakes area of New York state are some of the most scenic and beautiful areas in the country.
And don't forget the race track, Watkins Glen International. The track offers camping within their infield and outside the track for as little as $40 from Friday morning through Monday morning (plus admission cost). It's "rough" camping (no hook-ups), but it's fun with always something to do and see. I go there every year for about 3 - 4 weekends, but stay clear of camping at the NASCAR race as it's so packed (sold out year after year) it's like living in a city. If I was young once again, maybe, but after years of camping during the Formula One race weekends, I'm getting too old for that stuff today.
I know I'm sounding like a PR person for Watkins Glen, but I've lived in the area all my life and I never have grown tired of it.
Here are a few links:
Watkins Glen/Corning, NY KOA Campground Big Rig RV Park
Ferenbaugh Campsites in Corning New York State
Clute Park Campground | Watkins Glen, NY
Watkins Glen State Park - NYS Parks, Recreation & Historic Preservation
Watkins Glen International - Watkins Glen International
Wine Trails of the Finger Lakes | Finger Lakes Wine Country