Here's the deal: If you use a public toilet for any reason, you need to wash your hands before you leave. Don't try and debate it, don't get defensive, just do it, or else you might be publicly shamed by this DIY alarm that lets everyone know your hands are probably a cesspool of disgusting.
There's a detailed breakdown on how to build your own sanitary alarm over on Make, which uses piezo sensors on the toilet and the sink to detect the flow of water and determine if the sink has been used after the toilet has been flushed. These are connected to an Arduino unit which is also connected to a door sensor that triggers an alarm if a toilet patron is trying to leave without washing up.
The simple system only works with single occupant toilets, so public toilets will still be filled with monsters who don't wash up afterwards. But if you go to all the trouble of implementing it at work and only one person learns proper toilet hygiene as a result, it was still totally worth the effort. [Make]