That's a good question. I haven't gotten beet stains on clothing before. This article makes it sound fairly easy to remove using bread, of all things (and a stain stick): http://housekeeping.about.com/od/laundry/a/stain_beets.htm

My fingers get stained when I work with beets. I find that washing my hands gets rid of about half the stain, then washing the dishes later takes care of the rest of it.

I'm going to dye some Easter eggs with beet juice today.