The Ohio River doesn't have shores so much as it has broken glass and driftwood with plastic stuck in it
But seriously, there are a few well-kept parks along the Ohio River in/near Cincinnati. None of them are somewhere you'd want to go swimming or anything like that, though. Mostly they consist of paved paths on which to walk your dogs. The better parks in/near Cincinnati are in the hills overlooking the valley (like Shawnee) or along the tributaries to the Ohio. Kayaking and canoeing are relatively popular on the Greater and Little Miami Rivers and the White Water river, and there are good parks along all of those.
To get to any great parks, I'd say you'd have to travel an hour north to the Yellow Springs area near Dayton, an hour and a half southwest to the Clifty Falls area near Madison, IN, three hours or so northeast to Hocking Hills near Athens, OH, or three hours or so southeast to Red River Gorge near Slade, KY. Out of those, I'd say Red River Gorge is the only one where you can truly get into "wilderness" area and backpack, camp, or simply hike without seeing people or man-made things. Red River Gorge has some truly stunning rock formations and geology, but it's quickly being ruined by careless campers and hikers. Almost every rock formation is absolutely covered in carvings, almost every established campsite is littered with broken glass and tree stumps from where people cut down trees to burn, and all of the overhangs/caves are stained from the smoke of illegal campfires that drove out the bat populations and ruined the Native American artwork present at some of them. It's still a beautiful place, but a lot of it is spoiled by selfish or ignorant people.