I'd recommend Winton Woods County Park. Hamilton County Park District - It's Great Outdoors! ®
I worked 30 years in the surrounding school district and it's a great park. Downtown is about 20-25 minutes away.
East of town is Steamboat Bend County Park (Part of Woodland Mounds County Park) It is 20 minutes east of town on U.S. 52. Easy access to the Red's ball park.
South of Cincinnati is Big Bone Lick State Park (KY) and is interesting with a small museum and lots to interest kids. Big Bone Lick
I hope you enjoy your stay in Cincinnati.
Big Bone Lick is about 30 minutes south of downtown.
There are no closer commercial campgrounds except Mobile Home parks that allow short time stays.