I'm not sure that merely breaking up the banks would be sufficient. In the crisis that broke in 2008, it wasn't just one bank that got into trouble, it was nearly all of them, large and small. But yes, it was the large "too big to fail" banks that got bailed out, adding both the ability and incentive for banks to get bigger.

That is, banks that should have failed in previous crisis but were bailed out live on to become bigger, and banks have incentive to get bigger (among other motivations) so they will qualify for bailouts in the future. The regulatory environment has directly lead to the rise of the mega-banks.

So, while I'm not in favor of financial intervention, given the system that has evolved, perhaps you're right: Perhaps the big banks do need to be broken up to minimize systematic risk. But once that's done, the government must seriously scale back it's regulation of and intervention in the financial markets. Radical changes would have to be made, including the dissolution of FDIC (and probably the Fed), or you'll just have a bunch of smaller banks taking the same risks (because of the incentives and safety nets in the system) receiving bailouts.

As for bankers still taking risks even without a safety net: Yes, you're right, but there would be several checks on this in a freer financial market. Bond holders and creditors should act as a risk limiter, and so would bank customers, but with bailouts and FDIC, neither creditors nor customers have sufficient incentive to stay away from the riskiest banks. In fact, they have incentive to go to riskier banks because they can get better returns (not so much in recent years for customers, since interest rates on savings are pretty much zero across the board).