How should doped silicon wafers be handled?

Most dopant's are extremely toxic so yes you need to use the proper facility PPE when handling wafers. That usually includes gloves and eye protection at a minimum. Your body is also a large source of contamination (like skin oils) so proper handling devices like wafers tweeters should be used. Ultra pure DI water should always be used as a rinse to reduce surface contamination.

 

Extremely carefully!! Although its mostly because you don't want to break it. I've done work in a semiconductor lab where we handled doped silicon wafers in the form of silicon detectors used in collider experiments, although these are doped with different concentrations compared to wafers for commercial purposes. The most commonly used dopants are boron or phosphorus, but they exist in concentrations that are too small to affect you biologically. They are also bound within the crystal lattice of the silicon. The silicon is usually passivated, so you wouldn't be touching the dopants, but it's not recommend as getting skin oil on them could affect them electrically. So, touching them with your bare hands is, while safe, not good for the electronics. As nsaspook said, you can clean it with deionized water.

When you say you are building a diode, do you mean you currently have plain silicon wafer?