I have to agree with everyone else, I have never had a batik fray and unravel as you describe. In fact batiks unravel less then any other cotton fabric I have worked with.
Once I was working with a gold lame on a christmas quilt. It was rayon I think and man that frayed exactly as you describe and was really slippery to work with. I swore off lame with that quilt. Luckily it was just a star embelishment on a log cabin christmas tree. It drove me to fits. I could only get it to cooperate by putting on fusible interface.