My recommendation is to try a few different stabilizers on an old t-shirt (cheapie used one from a thrift store) to see which one you like best. Many like fusible tricot, but there may be a featherweight fusible that is better, and I personally would want to try MistyFuse (using a Teflon-like applique sheet to apply, as it does not have a paper backing).
There was a recent thread on this topic here:
http://www.quiltingboard.com/t-89167-1.htm . Some of the posts give specific names for the featherweight fusible, etc.
Oh, and I would consider using MistyFuse or one of the other fusibles on the bandanas before cutting into binding strips. I think it would look terrific as a binding, but the binding gets the roughest wear of the entire quilt so ultra-thin bandanas would really require some reinforcement with a fusible.