No, not really. It's not a massive issue for Oyster as the best you could do would be to clone a card and use it that day - it would get picked up overnight in the consolidation runs (if not before, they have some rather clever pattern detection software). They have other checks in place that make it tricky.
Mifare Classic cards are compromised now. It's not trivial. They do however carry various sectors that can be used and if what you put in there is encrypted then it's not a lot of use without the key to the encryption.
Got a modern smartphone with NFC built it? You can read a mifare card. Plenty of tools around to crack it. Ebay will supply all the tools and blank cards. It's pretty simple.
As pointed out, Mifare is a brand though - they have many other cards that are not (yet) compromised. Mifare DESFire is common (although IIRC, the original version of those have been broken).
Darren