What is the Source of the Problems?
The standard explanation for the problems in some of the countries, e.g. Greece, is that lack of effective monitoring of government deficits within euro area countries and lack of enforcement of the rules on how much debt a country can have allowed excessive debt levels to accumulate. In other cases such as Spain, the problem wasn't irresponsible budget behavior, it was the recession that caused the government budget to collapse.
Thus, the problems were generated both by bad behavior and by bad luck, and once these countries got into trouble, the deficit problems were made worse by the fact that countries within the euro area do not have the ability to use independent monetary policy. If these countries had their own currency, they could devalue and stimulate exports and this would offset the negative effects from raising taxes or cutting spending to address the deficit problem, but that option is not available.
What are the Possible Solutions?
There are two problems to address, how to prevent debt problems from happening in the first place, and what to do if problems develop despite attempts to prevent them.
As for preventing these problems from happening in the first place, one way to do this is to establish a bail out or insurance fund with the contributions for individual countries linked to their compliance with the rules regarding permissible debt loads (e.g. a debt to GDP ratio no larger than 60%).
A key factor here is the ability to measure debt loads accurately. Accounting tricks such as those used by Greece to hide its growing indebtedness should not be allowed. Thus, in addition to the need for penalties when countries are out of compliance, the reporting rules need to be strengthened so that compliance can be accurately assessed.
The insurance fund described above with contributions that are linked to a country's debt position is not the only way to dissuade countries from pursuing high debt strategies. But whatever strategy is chosen, countries must face penalties of some sort in order to change their behavior. If they know they will be bailed out if things fall apart and that no penalties will be assessed, there's no reason to avoidy risky high debt strategies. But, again, this requires a means to effectively monitor existing debt loads and the existence of a central authority with the powers to do something when countries endanger other countries through there sovereign debt choices.
What should be done if problems develop despite attempts to prevent them? One option is to do nothing and let the countries suffer a long period of high unemployment while they pay off their debt. The problem with this is that it allows the unemployed to suffer needlessly -- they could be helped but the choice is to turn away -- and the possibility of default remains which affects other countries through its impact on financial markets.
If, rather than doing nothing, the decision is to step in an help, there are several options available to help countries through their difficulties.
First, other countries could grant the countries having troubles short-term loans and credits, but this simply transfers debt from the present to the future and hence does not get at the underlying problem.
Second, there could be a direct bailout from other countries, i.e. the other countries pay off debts without asking for repayment in the future, but this sets up moral hazard problems. One way around this is to force countries to allow their budget to be controlled by a central authority in return for paying off the debts, but for many countries simply defaulting and leaving the EU would be a more attractive option. In any case, it's a bad idea to pay off the loans without penalty, so some sort of "conditionality" would be necessary. However, conditionality would be resisted by sovereign governments - it means giving up control of the domestic fiscal budget or other similar measures - and that is something countries do not want to do.
Another solution is to allow countries to temporarily drop the euro and adopt their own currency (in return for some conditions), devalue, and then return to the euro. However, the administrative problems with this proposal make it infeasible.
If the inability to devalue is the problem, is there some way for countries to devalue without temporarily going off the euro? Wage and price controls accomplished through a government decree to lower all wages and prices would work in theory, but the administrative and enforcement problems would be insurmountable, so this is impractical.
The final option is the one discussed above, an insurance fund of some sort that can be used to wind down countries in trouble. A fund of this type can also be used to blunt threats to default or leave the union that currently give countries in trouble leverage over other countries, so their are multiple advantages. The proposal to from a European Monetary Fund discussed here is a good starting point for the construction of an institution with these capabilities.
This institution serves two purposes. By linking the contributions to the likelihood of default, countries are discouraged from pursuing high debt strategies. And if troubles develop anyway, the resources are available to effectively deal with the problem without creating a systemic threat. I'm not sure if the politics within the euro area would allow such an institution to be created, I doubt that they would, but that doesn't mean it isn't a good idea or that we shouldn't try to create such a facility.
Finally, the European Monetary Fund would be an effective means of addressing monetary and financial issues, but the problems posed by a currency union go beyond the monetary realm. As I explain in more detail here, effective institutions for redistributing resources across countries so as to stabilize and enhance economic activity do not exist within the euro area. If a central authority is to be created to deal with monetary issues, consideration should also be given to "fiscal federalism" that will allow taxes and transfers among countries to be used to stabilize individual economies. The political problems are difficult here as well, but this should still be part of discussions about what types of changes are needed going forward.
With all of these proposals there is a tension between centralized authority and individual autonomy for countries within the union. Preserving as much autonomy as possible has been a guiding principle in the construction of institutions within the European Union, and its a good principle, but I can't think of any way to solve this problem without sacrificing some of the independence individual countries now enjoy.