I do both...on the same quilt. I cut the side borders on the lengthwise grain and the top and bottom borders on the crosswise grain. My reason is this: once they're sewn on the quilt, the grain all runs the same way, lengthwise from top to bottom.
What it means is the stretch is all going the same way; any fabric print is all going the same way; the light reflects off the warp and weft the same way which eliminates that subtle color difference that happens when the grain direction differs; and you never pull on the stretchy crosswise grain when you pull the quilt up under your chin on a cold night (or hang it on a wall).
It's just something I started doing many years ago because it seemed to make sense to me.
