(This answer will be in the context of 15th-century history.)
Simply, more exposure to diseases from animals means better immunity in the long run.
As Europeans were exposed to these diseases, they will likely have issues at first. But, their bodies will develop immunity towards these diseases to the point where many bacteria cannot even affect them at all.
Let's look at a civilization that mostly did not have livestock, Native Americans. They were pretty clean and healthy. That is because there were no diseases to kill them until the Europeans came to North America. For example, the flu virus was something a European could handle with a little bit of rest. On the other hand, the flu virus wiped out most of the Native American population since they never had exposure to it.