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I teach martial arts, and my wife teaches dance. I know of one student for sure (And I would presume that there are at least a few others) that have not received the normal United States progression of childhood inoculations.
I also have a <6 month old newborn that occasionally gets taken to the classes with us as we have somewhat conflicting schedules.
How much (if any) do I need to limit the exposure of the newborn to children that are not vaccinated?
There seems to be a link (as one would expect) between areas with a higher concentration of unvaccinated individuals and disease outbreaks (see this article and the source data map).
The recent (2015) measles outbreaks in California are a good example: at least 70-80% of those infected were unvaccinated (including a number of newborns too young to be vaccinated).
State officials say that 28 were not vaccinated at all, one was partially vaccinated and five were fully vaccinated. Six of the unvaccinated were babies, too young to be vaccinated. (source)
Simply put, unvaccinated individuals are more likely to be carrying a vaccine-preventable disease compared to the rest of the population. If your newborn is not vaccinated yet he or she is at risk. In the case of measles or whooping cough, an infection could turn deadly, as can many others.
I'm not an expert in pathology, nor am I a parent, but I would at the very least not let these individuals hold or come near your newborn, and would preferably keep my newborn completely away from them until a later time.
If you are immunized, there is of course less risk of you contracting a given disease through contagion from an of an unvaccinated carrier. However, if your child is young enough (e.g. < 6 months old as in the question here), your child is unlikely to have had the full range of recommended vaccinations yet. (The American Academy of Pediatrics has a list online with the recommended schedule.)
Common sense tells me that this is a real risk that is worth consulting with a doctor about. Keep in mind that you also need to find out about the vaccination status of the environments that you might choose as alternatives, like a baby sitter or day care.
I think you should find out from a doctor what the specific risk of contagion is like for the diseases covered by MMR, for example (that is one that is later in the schedule than the age of your child). Is it ok as long as no students handle the baby, or is the risk airborne or transferrable from them to you by skin contact (which could be a part of the instruction in either dance or martial arts)?
