Some rides you can just take them with you. On others the CM at the ride will hold them for you until you get off the ride.
This. In rides where the vehicle is wide enough, like PotC or Living with the Land, I laid my crutches along the floor (after asking everyone in the row if they minded, and nobody ever did). In rides with narrower, taller vehicles like the HM or Peter Pan, where they wouldn;t block anyone's view, I held them upright beside me. In rides where I couldn't take them on, a CM took them when I boarded and handed them back when I got off.
One time on the HM, I couldn't get my backpack off and my crutches tucked in beside me fast enough, and the lap bar came down over my crutches and couldn't lock, and the whole ride stopped! My sister said, "Nice going! You broke the Haunted Mansion!"
At Space Mountain, I asked the CM, "Is there somewhere I can leave my crutches?" I really meant, "Where do I leave my crutches," but I didn't word it quite right. He said, "Nah, just take them with you. They'll fly out of the car and land at the bottom, and you can get them when you get down there." Then he smiled, laughed, and said, "Just kidding; I'll take them for you."