We're helping you develop a rule of your world: and you have two options
Merepeople are an evolutionary divergence from humans. Therefore, just as humans are similar enough to some primates that disease can move back and forth between them.1, 2
Merepeople may look suspiciously like humans, but they're actually an example of parallel evolution. In this case, despite sharing characteristics with humans, they're not actually human or even primates. As a result, they're much better protected by the infamous blood-brain barrier and far, far less likely to become ill due to contact with humans.
Now, having said this, nothing's perfect. If I remember my biology correctly, nearly all viruses that afflict humanity began in birds. But very, very few of the viruses that affect birds end up affecting humans. This is because the illness must jump that nasty blood-brain barrier and almost always does so through a series of intermediary species (like pigs... e.g., swine flu) that allow the virus to mutate to eventually attack us.
If you follow #2, your merfolk have the same basic protection that humans have from all the really nasty illnesses that afflict the animal world. In this case, it would take a lot of contact with humanity (and the creatures humans keep with them) to eventually suffer an illness.
I believe those are your two options. If they're human mutants (or humans are merfolk mutants...), then disease would be reasonably easy and only a little contact with humans would lead to disease. If they're actually another species, then it would take a whomping lot of contact with humans to contract a human disease. But in either case, given enough contact, eventually somebody's going to get sick.
But, before I jump away, let's talk about saline
You could concern yourself with one thing: salt in the water. Illness (e.g., bacterium, viruses, etc.) like to feed on, well... food. Sugar is food. Salt isn't. It takes very little salt to kill something. Don't believe me? Drink a gallon of sea water in a day and see how you feel.
Your merefolk live in an environment that is naturally condusive to not being human-sick. This is an advantage — but it's only an advantage. Mother Nature tends to like throwing curve balls. Therefore, it is my belief that it's impossible for merefolk to be immune to human illness.
But I think it's unlikely they'd ever contract a human illness. It would take a lot of contact, less for #1 than for #2, but a lot nonetheless.