After you get it as clean as you can do you think it would be possible to use grout paint to paint it whatever color it should be? I have used calk paint on older tile floors and it works pretty well.

Photo? As Virginia Lynn suggests, removing the caulk and then painting may work very well. I'd try a satin enamel on an inch or so to see if it fills in the porous tile nicely.

I don't have tile baseboards, but my painters finished my wood baseboards by mixing the wall paint into some kind of caulk, which left this wonderful seal on the top of the wood that is easy to clean. I don't know the products used or the method they used (sorry), but maybe some research would lead you to that solution? I found this product through google, not sure if this is it or not:

http://www.sashco.com/products/exact-color/

Dust sticks to some caulk and makes it almost impossible to clean. Clean it as best as you can, then paint over it.