I've also discovered the truism that my chicken-owning friends have told me - if you get chickens, you're getting mice too.
What I've done since noticing droppings in my hanging feeder is to put a cover on the feeder. After the mice completed ignored the cover and slipped in anyway, I placed a rock on the cover. That seems to have done the trick - no dropping in the feeder since I've done that.
I'd empty out the feeder, or at least pick out all the mouse droppings, then fashion a secure lid...
Other than that, I guess either get a mouser cat, try and trap the mice, or do your best to limit what food the mice have access too. We've opted for the third option...if the mice get worse, I'll start trapping them.