You don’t need a license to operate a boat. There are age limits and education requirements — depending on the type of boat and how it is propelled — but no real “boating license.” Having a drivers license (for driving a car) has nothing to do with being capable of, or legally allowed to, operate a boat.
Ohio law requires any boat operator born on or after January 1, 1982 to successfully complete a boating education course approved by the National Association of State Boating Law Administrators (NASBLA) or proficiency test before operating any craft powered by greater than 10 horsepower. Typically when you pass a boating course, you get a card or certificate that would serve as your proof of education.
Aside from boaters who were born before 1982, boaters who carry valid merchant mariner credentials — issued by the US Coast Guard with at least one endorsement of master or operator — are exempt from Ohio’s mandatory education law. Documentation of these credentials must be carried onboard (as of July 10, 2013).
If you are renting a boat, the boat rental agency may have its own rules, above and beyond Ohio’s education law. These rules are usually dictated by their liability insurance.