Up to half of men using a public toilet do not wash their hands. Some of them fake it, because they feel enough social pressure to appear like they're not filthy, but not enough pressure to actually use soap. For this reason, I recommend the following options for public toilet designs that do not depend on changing the human nature of people who just do not care.
Option 1, noting that door handles will pick up germs, is to move the wash basin outside the toilet doors completely. This has two huge advantages. One, as implied above, it means handwashing occurs after using the door handle, which is obviously the preferred order of events (assuming there is a door handle at all, see below). Two, because handwashing is now occurring in public, non-hand-washers are forced to parade their behaviour under the withering scorn of everyone around.
Option 2 is to arrange the doors so that "Push" is an option to get out, allowing a clean human to use shoulders, feet, back, hip or elbow to push through the door, preserving the cleanliness of his hands until the next handshake or nose blow or computer keyboard use. The doors don't have to push only one way, either, and they don't have to be ordinary doors. Automatic sliding doors, two-way swinging doors, automatic revolving doors or even S-bend hallways with no doors would be acceptable. I have seen many of those options work.
A side-option, for the truly germophobic is automatic taps. Using a dirty hand to turn on the water, washing, then touching the dirty tap to stop the water is also considered futile by true germophobes, so automatic taps are a nice touch to consider, but not essential.
Mokalus of Borg
PS - You can turn off the tap with your paper towel as an option when the taps are manual.
PPS - The easiest changes to an existing arrangement are probably to have the doors swing outwards.