The best way to take care of a spindle is to keep it warmly dressed in wool, silk, or cotton ... LOL
But that's probably not what you mean. In terms of waxing or oiling, if the spindle whorl is shiny, it is very likely that it has a light lacquer on it -- if you wax it, the wax can't penetrate the lacquer, typically. I usually test an inconspicuous spot (the underside of the whorl) if I'm not sure -- if it's slippery ("slickery" is more descriptive, but not a real word...) then it's lacquered and all it needs is a good dusting.
Spindle shafts seem to be 50/50 (in my collection anyway) between having some finishing and simply being sanded wood. I used to "treat" my shafts with Wood Beams, but find I like the natural wood surface better for grippability both of the fiber and of my hand when I get the spindle twirling (finger-flick or palm-against-thigh). If you prefer a really smooth finish, Wood Beams does a very nice job on raw-wood shafts like those on the Bosworth spindles.
That said, I've had my fair share of water accidents (mostly with the niddy-noddy...) and find that a nice wood wax helps restore most of the shine and make it smell really nice. I use Wood Beams, Howard's Feed-N-Wax, or Citrus Shield from Howard's, depending on my mood and the scent desired at the end of the process ;-)
I'd say the main care I've taken is in keeping the spindles upright in spindle stands, to help prevent the possibility of the shaft warping. Though for some spindles (my Russians come to mind immediately...) that just isn't feasible. I did "score" a really nice skinny rectangular vase last Valentine's Day (ok, it had roses in it at the time!) that is terrific since its skinniest part is about as wide as the widest part of the Russians; so they are more upright than they've been in the past.
Do you have spindle care tips? Post your ideas in a comment on the blog or contact me. Thanks!