One of my favorite numbers guys, Nate Silver, did a rundown of all the governorships up for grabs this fall.  Of the Democrat-held states, 7 could realistically be flipped. Amongst states currently boasting a Republican governor, 8 are considered completely locked up.  Eighteen states with Republican governors bear some likelihood of a change in party.
Ohio is on the latter list; although according to the Five Thirty Eight analysis, which cites the Fallon Research Poll conducted by the 1984 Society in January, the current estimation would be that the Governor’s Mansion in the Buckeye state will stay Republican.  The article notes three leading nonpartisan analytics groups that anticipate Ohio’s Republican lean will continue, with a clear nominee from each side having already emerged from what they noted was an “active invisible primary” over the winter.
How the “blue” wave that is forecasted in November will affect other statewide and down ballot races is too early to tell, but projections for Ohio’s Governor’s race remains consistent.
You can check out analysis for the rest of the contested states here.