Anti-Catholic sentiment + the state was traditionally Republican at the time
I doubt whatever anti-Catholic sentiment there was in Ohio was greater than in the nation as a whole given the long history of Catholics in the state (typically of the middle class German Republican variety). It is true that the politics of Ohio were fairly Republican and conservative then. If you compare the difference between the national and state margin, it's not far off from what it had been in 56.
Something along these lines. German Catholics, unlike Catholics of most other ethnicities, weren't necessarily Democrats in that era even when it came to JFK. Purely anecdotal, but a decent fraction of my German Catholic ancestors still voted against JFK. The result map bears this theory out somewhat:
Nixon won Putnam and Mercer Counties, albeit barely, both of which are pushing 70% Catholic today and probably higher back then (the German Catholic settlers of those areas came when they built the canals; long before 1960). JFK probably did better than a Generic D among German Catholics, hence the shade of blue of the counties in NW OH are clearly related to the % Catholic. But he certainly wasn't pulling 80% and probably even lost Ohio's German Catholic vote.
Also notice JFK's loss of heavily German Catholic Hamilton County. (Granted, the other demographic groups there wouldn't have been friendly to him)
The Polish/Irish/Italian Catholics in Cleveland are of course a different story.
