I am going to frame challenge this question.
Why is this a problem? Are the aliens sexually compatible with humans? Can we see human/alien hybrids?
Or perhaps their alien-cows can breed with human cows.
Life is hard, it requires very specific conditions for it to function and flourish. Give a virus/bacteria too much heat, or to little it will die2. Mixup the PH of the substrate: too low or too high, it will die.
Take a fish virus, and stick it into a human? The virus dies, the human doesn't notice it, human cells don't have the mechanisms that virus exploits to duplicate itself.
Fish bacterial infection? Humans run too hot, it dies.
Then handwave the problem away with incompatible biology1.
If a cross over event did happen, it would probably be extremely lethal and kill a lot of humans, possible all of them. And human scientists without any past experience with the new diseases wouldn't have enough knowledge about the disease to make a cure/vaccination. Aliens wouldn't have any knowledge either, being unfamiliar with human biology...
So don't have it be a problem
The chance of a virus that infects a species completely unrelated to us is very low. Their cells just wouldn't have the right mechanisms for a virus to hijack, and alien bacteria wouldn't be adapted to their biochemistry. Just think how rare of an occurrence it is for a virus/bacteria to jump hosts with creatures that are pretty close to us (Cows, Rats, Dogs, Cats...)
 There might be a crossover event in the future as "Life, Uh, Finds a Way." There is a chance however slim that a virus/bacteria mutates something that allows it to survive in both Alien organism of some sort and Humans, but that is a problem for future humanity, not current humanity. It also might never happen, if humans/aliens are different enough biologically. I don't know of any instance where diseases jump from Alligators to Humans, and we come from the same common ancestor.
 Extremophile bacteria exist that can survive in both really hot/really cold places, However those extremophiles do not survive in places where other bacteria live, they only seem to survive/thrive in environments hostile to everything else.