How to re-anchor banister to plaster wall?
Move the anchors to a stud and attach them with long screws. Use a stud finder (preferably with a deep scan capability since you mentioned plaster) to locate your studs. Or you can drill some small test holes since you already need to patch the wall.

Don't use drywall screws, you need something with more shear strength. I wouldn't use anything less than a 2" screw, particularly in the top two holes. You need it to have a good bite into the stud.

Then you just have to patch and paint the old location.