You formed a union with your neighbors to work within a single financial system and a single currency. You are all mismanaging it together so you should all pay together. Not only because you all caused the problem together (so you deserve to share the loss), but because if any of you go down you all go down. You don't have to like each other, you just have to keep each other afloat when things get bad. If you didn't want to pay for things like this then you should not have set up such an insane arrangement in the first place.
The above might be an argument to bail out states. And actually; I dont mind too much if some of my tax money went to greece or spain or cyprus in the form of debt relief (which makes a lot more sense to me than lending them even more money they cant pay back). Its also in my interest those states get back on their feet financially and economically.
But the above can not possibly be an argument to bail out failed banks that took ridiculous risks and lost it all. The moral hazard is exactly the same as in the US bailout; which I think was a colossal mistake. Those banks and its shareholders should have been wiped out so that banks (and their shareholders) that didnt mismanage their customers fund would actually have had benefit from their better strategy. Im glad the EU is doing this part right at least.