According to the research I did, organically managed dairy cows get less infections to start out with, because they aren't being fed all the hormones and stuff. When they do get sick, most organic farms will just sell animals that need to be treated with antibiotics to non-organic farms. It makes sense to me that they wouldn't want sick cows hanging around making the rest of their cows sick. I'm sure there are probably at least some organic farms that are not as good as others. I also know nothing first hand about the subject, so I could be entirely wrong.
One thing I do know first hand is that the milk tastes better and lasts longer without spoiling.