When I had to do this to a medical standard, so I could give my partner IV antibiotics at home, the official method I was taught was to use a clean paper towel to turn the tap ("faucet" for merkins) on and off. Elbow or possibly paper towel again for door handles.Did you clean the sink [especially the handles] before and after using it to wash you hands?
And then there's the part about touching the door handle.
In normal life I use my sleeve for public door handles. For taps in public bathrooms, hopefully they're automatic, or designed so you can use an elbow or something.