One important thing to note is that this sort of organization is only possible for ideologically motivated groups. If Anonymous existed for profit, they would once again become vulnerable. Money leaves a trail that the authorities can follow. Money changing hands creates the need for accounting, so that the various members all know that they're getting their fair share. If Anonymous went from protesting to pilfering, the change in their behavior would destroy the group faster than any law enforcement agency could.
However, the open nature of the group means that anyone can be (or pretend to be) a member. An FBI agent can show up in their chat room and talk to them, but he's not going to learn who they are, or where he can find them.
The point is I don't think Anonymous is suddenly going after credit cards. Extra Credits pointed out that it goes against the group ideals. I'd also say it goes against the machinery of the group. If Anonymous was in the business of stealing money, they would have been rounded up ages ago. Barring a truly foolproof way of un-traceably transferring money over national borders that allows both the giver and receiver to exchange money in subpoena-proof anonymity, it would be impossible for them to turn a profit.
It's been suggested that while not condoned by the group itself, the PSN attack may have been perpetrated by some subset of Anonymous. It's possible, but un-provable and sort of beside the point. Anonymous does a lot of work that focuses on counter-security. Certainly some people are attracted to the group because they enjoy breaking into things, not because they are honestly interested in everything the group stands for. (This is allowing for the fact that it's hard for an anonymous group to definitively "stand for" anything.) It's also reasonable to think that some of those people might do some non-Anonymous type hacking in their free time.
I don't think Anonymous is nearly as dangerous as people imagine them to be. Yes, they hack websites and interfere with the normal operation of the web, but is that really worse than other types of protesters? Think of the picket lines that block government buildings, shut down companies, or blockade traffic during rush hour. Those are annoying if they get between you and where you're going, but they're also a natural side-effect of living in a society where people are free to protest peacefully. This isn't to say that their actions are inconsequential or should go un-punished, but that we shouldn't blow them out of proportion.
Secret societies have always been unstable things. It's hard to make an impact on the world without being visible yourself, and making yourself visible makes you a target to people who oppose your ideas. It's an interesting dynamic, and I can't think of a similar situation in history.