What is a person called that explores caves?

A cave explorer is a Spelunker. NO: a caver.

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This has come back round. so with new knowledge I am re-writing my answer to explain why "No".

The Word "spelunker" was coined by an American caving club in the 1940s (according to an entry on Wikipedia which I believe to have been written by a caver in the US), from Greek and Latin Words for cave. It was used happily until the 1960s, but only in the US, then American cavers themselves started to see and use it as a derogatory term for ill-equipped would-be cavers.

Apparently some US cavers even wore T-shirts with slogans like "Cavers rescue spleunkers" - rather forgetting that they, the wearers, were beginners themselves once!

The proper term is "caver" - whether exploring caves or simply visiting known ones - or "speleologist" if studying the caves and their contents scientifically. Even then the individual is more likely to referred to him/her-self as a "caver" and leave the Word "speleology" to name the cave sciences in general.