A mouse can get in a hole the size of a nickel and sometimes even smaller. It would be pretty hard to seal up all the entrances, but you can surely try. Get a couple of tubes of caulk and steel wool and begin crawling around your foundation and closing up all the mouse doors.
Now, on the not so good side, maybe these critters never came in from the outside. Maybe their home is yours also. One will never know this.
Begin using the conventional traps and when setting and touching these traps, make sure your hand is inside of a plastic grocery bag. You do not want your scent on the bait or trap itself.
Where the siding over laps the walls of the basement's exterior walls, check for large gaps. clear away branches or limbs that these guys can use as bridges or ladders to gain access to your home.
You have a tough challenge ahead of you and you may not win, but hang in there.