In my opinion the biggest economic objection is the fact that it will be deflationary, and there's really no getting around this. As much as people complain about having a federal reserve, having an unmanaged fixed-supply currency is worse.
As far as criminal uses, IMO the jury is still out on that. Right now, it is popular for things like drug trades because it is an easy way to transfer money anonymously over the internet without involving a bank. Also, unlike a bank account, there is no way for any government to freeze or seize assets in a bitcoin wallet. But I don't think there are any large-scale criminal enterprises using bitcoins a major means of transferring money (of course, if they were I wouldn't know). Right now, most bitcoin users are just using them as speculative investments, and only a few small time drug dealers or libertarians are actually using them to transact commerce.
One reason why bitcoins might not be the criminal free-for-all that some people fear is that all bitcoin transactions are recorded, so one could argue that they are more traceable than paper currency. The problem is that the transactions only involve ID numbers and not names, so unless the government can figure out who owns which bitcoin wallet, then the bitcoin economy remains anonymous. And bitcoin wallets can be created anonymously at any time. On the other hand, if the government suspects that a certain bitcoin wallet is being used for drugs, say, they can wait until the owner tries to either convert the bitcoins to currency or spends them in another way that reveals the user's identity.
This leads to the issue of bitcoin laundering. Right now, it is pretty easy to launder small amounts of bitcoins. In fact there are websites that will do this: you transfer some bitcoins to them, and they transfer back the same amount of bitcoins minus a fee from a different wallet so that now you have "clean" bitcoins that aren't connected by transaction chains to your previous wallet. But it's still pretty small-time, and it's not clear whether this would scale to the point where large criminal enterprises would be able to launder huge amounts of money regularly, or whether the fact that there are transaction records actually would end up benefitting the police more than the money launderers.
= new reply since forum marked as read