What is the best way to finish a beaded braid (Bo Derek style) once I
have finished the thin braid and added the beads?  How do I get it to
stay in (fine hair)?

Put colored beads on the ends or add extra hair.

This little rubber bands that are given to kids with braces are the
best for ending little braids in even the finest hair.  And they come
in all kinds of bright colors too.

I have very fine, curly hair. It's difficult for me to keep anything
in my hair (bobby pins, barrettes, beads, whatever). My suggestion is
that you use a mousse or gel to add body to your hair before braiding
it. This will also help with "stragglers" that tend to poke out from
the braids.