they check... a lot, at least in my experience.
Two times I rode without a ticket (once hiking up and once sleeping in a ravine) were the only two days I was never checked. I've seen them intercept many people hiking up with skis or board, or they'll just cruise through the park and ask to see your ticket, or you'll just have to show it on the mile or palmer.
As for when to go, late June into July is usually prime. Right now, for instance, the weather blows, even though it is the beginning of June, Make your drive worthwhile and just camp at the airstrip for a couple of days.