Cleveland Lakefront State Park is open year-round and has a playground with equipment, a reservable picnic shelter and lots of tables and grills out on the lawn.
A lovely, tree-lined walkway parallels the swimming beach and connects the park with Villa Angela Park just to the East of Cleveland Lakefront State Park, so leave time to stroll and explore.
http://www.stateparks.com/cleveland_lakefront.html
8701 Lake Shore Blvd, Cleveland, OH 44108-1069 Cleveland, OH 44108