Replacing Louvers on Shutters

A hobby/sewing store will have lots of trims to choose from. The one usually used on furniture is called "gimp,", a sort of braid. It might be too wide or thick for your purpose, so look for a flat trim in an appropriate width.
When stapling the fabric, try to get the staples in a straight line so it will be easier to cover all with one strip of trim.
I recommend wood glue, but thin brads would work if you are careful not to stretch the trim or let it buckle. Also, don't try to hammer a brad into a hidden staple!

I once recovered a stool and used braid to hide the staples. You'll get it from the soft furnishing department in store. You can tap in small pins between the threads of the braid to hold in place, alternatively use glue.