I do it by using a medium-width zig-zag stitch. I hold the line against the fabric, and zig-zag over it.
Another idea is to avoid the fishing line entirely and just make a rolled edge hem. This is easiest on a serger, and on a serger you can go around twice to provide a little extra stiffness. Fishing line is kind of out of fashion right now (although it might be right for whatever you're making, obviously I don't know the details).