You people crack me up. A boarder is a person who eats at your house, and may have a room there as well. You would be giving them "room and board" A border is something around the outside, as in: Lake Superior is one border of Wisconsin. So the thing you put around your quilt is a border, not a person.
Sorry about that! I couldn't help myself. Once a teacher, we're always ready with the red pencil!