Cantelope:
Depends which model of Asics you are talking about.
If you mean a court shoe models are rocket, blade, resolution etc, then these would be fine. They have a gum rubber sole (very soft compound) for ultimate grip and are really designed for volleyball, squash, badminton, table tennis etc but would be fine on the basketball court. They would be much lower cut and hence lighter than a standard basketball boot, and as they are designed for court sports have a good longitudinal rigidity which provides lateral support for side to side movements etc. Asics are considered a soft shoe so the cushioning aspect would be fine.
If you are talking about a cross trainer, models such as the 140TR, 480TR, 580TR etc, then most of these will be fine as well. These shoes are designed as a multi-purpose shoe and most have a herringbone grip (wavey lines of grip) which provides good grip on smooth surfaces ie. Basketball courts. I would look for one that has rear and forefoot gel ie. the 480 model as some cross trainers are more geared towards a walking shoe which is designed with most of the cushioning in the heel. Cross trainers are also fairly rigid so will provide good lateral support.
Running shoes on the other hand are designed for straight line running, and have a blown rubber or carbon rubber outer sole in a pattern designed to give flex and cushioning in the right spots of the when you are running. Asics models include Kayano, Nimbus, 2130 etc etc These shoes tend to be more flexible and offer little lateral support, are probably too soft for the basketball court and if they have a carbon rubber outer sole will also mark the court. Although in saying that quite a few female basketballers and even the Australian netball team get away with wearing them, most probably because they are lighter and tape/brace their ankles.