Our cat is free to go in/out through the cat flap. Being quite wet here, he often gets muddy paws, so we get prints in the kitchen, carpeted rooms, window sills, today even on a wall!
On the inside of the cat flap, there's a mat, although that doesn't help much (and searching the web, everyone says so). On the outside decking, then grass / raised beds.
What we do currently
Currently we deal with it by trying to catch the cat (not that hard, if we're around) as soon as he enters, then wiping the paws (he's annoyed by that, but it's possible). This gets rather annoying near meal times when the cat is going in/out all the time.
Another thing we do is to close the cat flap when it's particularly wet, or just when he can't decide where he wants to be.
Any other suggestions? On one forum I saw a post mentioning in passing they have their cat trained to wait to get wiped after coming in. Not sure if that's even a real option.