Simple questions with oh so difficult answers. Indeed, it took a decision by the United States Supreme Court in 1980 to set the border between the States:
The short answer, as announced by the US Supreme Court, is that the border between Ohio and Kentucky, as well as Indiana and Kentucky, is the low water mark on the northerly shore, as the river existed in 1792 (the year Kentucky entered the Union). OUCH!!! Much easier written than found.
With the building of the dams and the corresponding substantial rise in water level, as well as the natural wandering of rivers, this "static" northerly location is buried beneath the flowing waters of the Ohio River. So while out kayaking, canoeing or boating in Kentucky, Indiana or Ohio you may wander and roam from State to State with nary a clue....HAPPY BOATING! (wherever you may be.....)