Mira is right. All holes & cracks need to be patched. If you can fit your pinky finger in it, a mouse can fit in it! Get some wire mesh & some tape, glue, staple gun (depending on if you're attaching it to concrete, wood, etc) and go around your foundation. Leave 1 spot open so that you can reach in & out with traps, etc.
Snap traps are still the best. Get the plastic kind. They're easier to reuse, hose off, etc. Peanut butter is the best bait I've ever used.
If you do get a cat, make sure it can't get to the traps - it could break it's poor little toes!
Cats can be a help, but I've noticed all my cats do is alert me to where they've seen a mouse. They never actually catch it, but my trap does every time!