If your bit fits your horse properly and you are not a "hand rider", there is no need for a curb or chin strap with a snaffle bit. A rider with good hands does not pull hard enough on the reins to pull the bit through the horse's mouth.
If your horse is opening his/her mouth, then the best route is to investigate the cause of that behaviour to change it. In the meantime, a full cheek snaffle bit may be the best option as the cheek bars give more stability and prevent the bit from being pulled through the horse's mouth. Because the bars also provide more lateral pressure to the corners of the horse's mouth, you may find that you don't need as much rein.
Ideally, the rider should be steering the horse from his/her seat and leg and using the reins only to provide boundaries that tell the horse where not to go. Many rider's aspire to be able to ride their horse's bridleless and the only way to get to that level is to ride less off the hands and more off the seat. Is your horse getting what he deserves or what he needs?