I check my tires after every ride. Sometimes I'll find and pick out a little piece of glass or flint, thus preventing a flat on my next ride. Personally, I hate tires that cut easily.
I've tried using urethane caulk and superglue to close tiny cuts, on the idea that leaving a cut will allow future debris to find a home and thus cause a flat. I don't think it does any good. In fact, I've had flats from superglue slivers. So I don't bother anymore. I just clean the cuts after a ride along with looking for newly lodged debris.
If a cut is large enough to cause a bulge in the tire carcass that I can feel with my hand, I toss the tire. I've tried booting over these things in the inside but that doesn't work, either. The problem is that the tube squirms in the tire and the edge of the boot eventually wears through the tube and causes a flat. And yes, the Park boot material will do this. If you can get a glued tube patch to stick to the inside of the tire, that sometimes works, but not all tire material will accept patches successfully. And riding on a tire with cut cords in the carcass gives me a bit of the willies. There's enough stuff to go wrong already.