Clostridium botulinum is a gram-positive, rod shaped bacterium that produces neurotoxins which can cause muscular paralysis. Ironically, it is also the paralytic agent in botox. It is an anaerobic spore-former commonly found in soil. As I said previously, heating destroys the neurotoxin but not the spores. Paradoxically, you can eat handfuls of the spores and it will have little or no effect. Acid in the digestive tract prevents the bacteria from multiplying. So Botulism is NOT a disease, it's a reaction to neurotoxins that are produced outside of your body. This is very different from E. coli where relatively few bacteria from contaminated food can rapidly multiply and become an internal infection.
Because botulism is not an actual disease process, it is not transferred from an infected animal. That does not mean that spores aren't passed along by the milk handler. The problem comes when Clostridium spores find a nice low oxygen, high pH (low acidity), nutritional environment.
Clostridium tyrobutyricum is of more concern to cheese makers. This is the one that generally causes “late-blowing” defects in high-pH cheeses such as Emmental or Gouda. Even low spore counts can bring about this phenomenon, if the growth conditions are suitable. So, what are the symptoms of late-blowing? Well, Clostridium produces butyric acid, acetic acid and hydrogen gas as the major fermentation products. So, irregular eyes, slits, and cracks caused by the production of the gas bubbles as well as an abnormal flavor from the Butyric acid and hydrogen compounds are a good clue that the cheese was contaminated. Unlike botulism, there is no lethal neurotoxin so it really won't make you sick. It will just taste disgusting.
Clostridium contamination happens on a regular basis with home canned foods even if you do everything right. If you open a "bad" jar of home canned beans, you will know it immediately from the gas bubbles and a strong hydrogen sulfide (rotten) smell. Same with a bad cheese. There will be no doubt.
I have had late-blowing happen twice. In both instances my wife let our dogs into the kitchen while I was making cheese.