It can be a couple of different things causing the light to stay on. The most common is a sensor inside the seat belt buckle that will fail. The repair would be to replace the seat belt and the tensioner for the seat belt. The cost varies from model to model. Get the car checked by a shop with the right equipment to be able to diagnose the propblem 1st. This is just one of the most common problems we see. There are many other things it could also be. Downside to the light being on is that the air bag will not function with the SRS light on, so it becomes a safety issue as well.
I have a black on black 5 speed 1995 525i that had this problem it turned out to be. When i put weight on the passenger seat books,bags etc the sensor in the seat makes this SRS stay on til you remove the weight from the seat or hook the safety belt(lol)then the SRS light goes off and stay off . Till you start the your BMW again . Just remember not to put weight on the passenger seat when starting your BMW.