We have stayed at Four Mile Creek (Niagara Falls), Hamlin Beach (30 mins west of Rochester) and Fair Haven (near Oswego). Large clean beaches, older bathrooms, otherwise they are typical state park setups...
All the state parks right on Lake Ontario (4mile, Golden Hill, Lakeside, Hamlin) do not have full hookup. Golden Hill has a few sites with water.
The closest site I know of near the lake, at least in WNY, is Wildwood KOA in Medina.
We have made Lake Bluff campground Lake Bluff Campground - Chimney Bluffs | Lake Ontario Campground on the east side of Sodus Bay one of our first trips every year. While it isn't on the shore of the Lake it is a fairly quick ride (less than 5 minutes) from the state park. We love going there and hiking thru Chimney Bluff State park with our 8 year old son. It is a nice area and we have had some good experiences with the owners and other campers there. I have heard great things about Hamlin Beach State Park but have yet to go there. I am actually considering going there this year. Another campground that is fairly close to the Lake is Webster Park Campground https://www2.monroecounty.gov/parks-webster.php which is run by the county. It is a very small campground but is nice to go to. Also it has a lot of hiking opportunities as well.