Do stray cats have diseases
Like any cats, stray cats can have diseases. In this sense the answer to the question is rather obvious.
When it comes to whether stray cats are more likely to carry diseases – to the extent that interaction with them expose humans to any measurable increased risk of catching a disease, the evidence suggest that this is not the case. Indeed, catch and neuter programs, where stray cats are vaccinated erase many of the differences between stray and non-stray cats. Moreover, people interacting with stray cats are not falling ill. The last confirmed case of a cat transmitting rabies was in the 1970s. In addition, parasites are often very host specific, i.e., they stick to, for example, cats, or humans.
Still there is a reason why cats are vaccinated (and all stray cats are not vaccinated), and there is scientific evidence that stray cats are more likely to carry diseases. However, an obviously ill cat will generally neither approach nor be approached by humans for social interaction. As with interaction will all animals, some common sense should be used when interacting with all cats.