I don't think it matters really. They say bias bindings hold up better - and if you're doing a scalloped border then a straight grain border won't be stretchy enough to go around the curves so you'll have to do bias binding, but for a regular square or rectangular quilt I think you're fine. I have quilts that have been washed a hundred times in the washer and tossed in the dryer afterwards and their straight grain bindings are still holding up strong. (Doesn't matter if it's lengthwise or selvage to selvage, it's still straight of grain.) I can't say that it will live a hundred years after me, but I won't be around to be heartbroken about it.
