I have done both. I determine the cut based on the type of fabric. If I use a thick backing, then I prefer bringing the backing to the front and doing a hem around it rather than binding. I posted a tutorial over in the tutorial section on how to do this without excess bulk on the corners. The thick backing (like minkee or fleece) is much sturdier than cotton binding.
I do have one quilt with frayed binding that I need to cover up. It's been much loved so I'm not sure if it's because of the straight of grain or just lots of use over the years. The ones that I have hemmed show no stress on the edges.
The worst that I have done was buying that ribbon binding for a baby quilt. After a few washings it looks horrible.