Considering the alien race to be just another species lurking on the face of the earth, only this one's ancestors don't trace back to the same roots, one could just as easily ask if they were suspectible to diseases carried by parrots.
It's unclear from your question how the aliens and humans are able to coexist. Is it because they are indistinguishable from humans, or because they are actually cattle and we just don't know they actually came from the planet Cowanus, or do they simply live amongst us as an acknowledged party of society? In any case, I think it's safe to assume that these aliens are unable to reproduce with humans, as they do not share... well... anything.
At the moment of arrival on earth, many of these "alien pioneers" probably did die to numerous diseases (viruses, bacteria, parasites, fungus...), but as always, a small amount of them survived. These aliens have, in the past 6000 years, evolved into a species that is accustomed with Earth's mother nature.
Another way to look at it is that they had the technology to come to this planet, from assumably very far away. They are hence likely to pack a bunch of medical resources as well, so just dying from everything that touches you is a bit much. Of course, this doesn't mean they aren't "suspectible" to these diseases, just that they're not immediately kicking the bucket.
So to answer you question, sure, they would be suspectible to human diseases. They would be equally suspectible to diseases carried by butterflies, gorillas and Lucius Malfoy.