Exercise is shown to reduce problem behaviors and increase on-task behavior. Sadly, those with autism are frequently left wandering and wondering in physical activities/fitness and PE (physical education). Children with special needs are still kids. They can play, run, swim, shoot baskets, kick a soccer ball, and play catch, just like their typical counterparts. They just need to be taught in an adapted fashion.
There is no magic answer to what type of activity is the most appropriate for children with ASD. The goal is simply to get the child moving.