@Boyan may already know all this, but for those of us who don't know much about this "tourist" cave dive, here is an informative article I found with a quick Googling:
Budapest Cave Diving Experience at the Molnar Janos Cave | The RTW Guys
"As required by strict European Union regulations, to dive inside the Molnar Janos Cave, you have to be cave dive-certified or be a student working towards your cave diving certification.
However, there is a loophole in this requirement that allows other divers who are not cave certified to do an introductory dive."
From the description, my guess is they have had a good safety record because they apparently vet the divers pretty well--probably better than in Mexico, where they allow just about any diver to go.