After washing your hands, use a paper towel to dry. Once you're done, use that same paper towel to clean the counter and sink and faucet. You'll always have a clean sink.
This is something that I've been doing for years, but instead of cleaning the counter & sink with it, I simply use it again to dry my hands the next time I wash them. (Which is SEVERAL times, while cooking a meal). Your hands are clean after you wash them, so it is okay to use it again, for the same purpose.