Basil goes brown and woody at the base. This is perfectly normal and not a problem. It doesn't need fixing.

But the roots seem to be exposed on your plants. Looks as if the soil has been washed away. I'd top up the soil (which looks too wet) and give them plenty of light and heat.

Thanks for the advice -- I'll definitely add more soil this weekend.

Another question, does the woody-stems process sometimes eat away at the side of the stems? It almost looks like they're decaying, since the stem gets thinner where the brown parts are. Is that part of the process?

Thanks again for the help!

(More pictures:
http://i1257.photobucket.com/albums/ii518/anhoffma/photo8.jpg
http://i1257.photobucket.com/albums/ii518/anhoffma/photo6.jpg )

Does the container have drainage? If not, likely that's the problem.