Wow, I didn't think this would as interesting a post as it has been - look at all the different suggestions! LOL* I never would have considered what some the other posters mentioned, how neat! Now I'm inspired. ;)
If it were me I would probaby go with a zig zag but with invisible thread. I don't care for the wide zig zag on my machine, I usually drop it down in size a little bit, it seems to look neater that way, at least with my machine. I love the idea of going in the ditch down the lines with a straight stitch for each of the colors. You could be sure then it would be nice and secure, and it would add a little dimension to your beach ball too.