My suggestions are Queensberry bay, Dumfries, and Brighouse Bay near Kirkcudbrigh, the first is right on the beach side, and quiet, the second is larger and busier, probably suited to familes, but is also by the sea. Check them out, you will not be disappointed.
I agree Brighouse Bay is close to a nice wee beach, just over 2 hours from Glasgow, my daughter's favourite. Yellowcraig is about an hour and a half, near North Berwick, again, a short path to the beach.(Bunroy, above Fort William, is over 3 hours away and Carradale Bay, on the Kintyre Peninsula also over 3 hours)