I imagine I would have a long talk with my neighbor, asking questions about the illness and any known facts about its transmission, and if the child freely plays with other kids, if the child has been trained to follow some basic hygiene rules, etc. Age would make a difference, the older the better....a 3 year old spreads around a lot more saliva than, say a 10 year old.
After getting information from the neighbor, I'd talk with my own child to explain why there is a need for greater hygiene precautions when playing with this neighbor, and I wouldn't allow my child to do so until I trusted that s/he would follow through.
If my concern level was heightened and/or my own child had a compromised immune system or was fighting a bug at the time, I might make face masks (used and worn properly) a condition of the play date. I'd have both kids wear them, so that neither one feels singled out and it'd also decrease potential opportunities of infection.
I would make sure to be completely honest with the neighbor about any concerns I had, but I would do everything I could to keep the relationship open both between parents and between kids. It's a great educational opportunity for them, and gives those neighbors a friendly place to turn to when everyone else is skrinking back and not lookign them in the eye.
If it reached the point where I was just too concerned, I would say no to a conventional play date, but would still encourage interaction, such as the kids talking with each other through the phone or accross the yard, etc. I very much doubt that I'd become that concerned, but if I did I would still try to make things work while doing what I felt I had to do to protect my kid(s)