Poor you, but it will get better. IME the first year working with small childen is the worst in terms of picking up illnesses. Your immune system kind of get to grips with it after that.
Many primary school teachers experience the same, especially when they just start the job.
I don't think you can ban snotty noses (unfortunately ...) as you won't have any mindees left. Little kids pick up stuff ALL THE TIME. Not much they can do about it, not much parents can do about it, either.
The only thing in your power are to keep the place as clean as possible, encourage good hand washing by supplying sufficient child friendly soap and paper towels (many small kids are terrified of hand dryers).
Very regular cleaning of toys (many can just go in the dishwasher, even lego, on the sterilisation program), other toys can go in sterilising bags with milton or washing machine (hot wash) or in freezer (but for quite a while I think, not just a few hours), also scrupulous hygiene re food to avoid tummy bugs.
I think it is a good idea to wash mindee's hands when they arrive, however, be careful, many kids have very sensitive skin, too much hand washing can cause all kind of problems, same for hand gels etc.
I use Vicks hand foam, as it stays active for four hours, but not sure if you can use it as you have to be very careful it does not come close to any eyes (it puts a thin layer on your hand, so stays on once applied and even once dried will still hurt terribly if touching a child's eye).