Physical Danger
The danger is almost entirely on the side of your patients (a.k.a. the test subjects). Sure, you're around all sorts of diseases. Some cell biologists actually spend their entire lives modeling one strain of one disease, with the belief that understanding how that disease progresses on a cellular level will help us eradicate it. (Or they're just morose, disease-obsessed people.)
By administering the stem cell therapy, you won't really come into contact with the infected cells yourself; they won't be in your bloodstream or joined to your existing heart muscles.
However, despite the existence of more guidelines, rules, and regulations than you'd ever wish upon your worst enemy, you'll worry constantly about the safety of your patients. Stem cells can be contaminated, meaning that you're introducing more of the exact cells the patient is trying to fight. Or, with a flawed donor screening process, an infected donor could get through the system (source).
When you get right down to it, it seems like "constant worry" is the biggest worry of all.