Alas the only way you’ll ski Kentucky these days is to find a hill (there are plenty), climb to the top and wait for a snowfall (not so plenty). Fortunately there are quite a few ski resorts within a reasonable drive in Ohio, Indiana, West Virginia, North Carolina, and of course Gatlinburg Tennessee.
Here is some information as far as skiing, snowboard, and the Commonwealth proper…
Lexington Ski & Sports Club — This fun group proves that in addition to horses, bluegrass, and bourbon, Kentuckians are also passionate about their winter sports.
Ski Butler (Lost) — Not in Butler, this ski area was in General Butler State Park in Carrollton. Built in 1982, this resort featured one triple chair lift, two double chair lifts and a rope tow on 25 skiable acres with a 600-ft vert. Aside from very little snow, Ski Butler had everything you’d expect at a modern ski resort. A quick look at the trail map above reveals it was an upside-down area (lodge at summit). It also looks like they had transfer tows at the bottom; not sure. Anyway the point is moot as Ski Butler shut down sometime in the mid to late 1990s.