What is jersey?
You can knit a different color border along with the scarf, but it requires you do use intarsia. With intarsia you'd have an extra two skeins of the color along with the two design colors for the middle. It's not impossible, but it would be awkward and certainly not portable.
It is possible to knit a border after the fact by picking up stitches along the sides. With a scarf you'd probably want to pick them up one side at a time and knit although if you used a really long cable you could do it all at once and miter the corners.
There is a thread in the How-to Forum at the top called "why is my stockinette curling" or something like that and it gives suggestions for non curling borders such as seed stitch and garter.
All this being said I will say though that adding a border after doesn't always help as well as you'd hope. It tends to want to fold along on the picked up stitches. I'm not sure how to combat this.