I think that it is safe to say that we no longer live in a world where things in one country happen in a vacuum and there are no other consequences for the rest of the nations of the world. Cyprus is a good example of such a problem, and other countries will be hurt.
The decision of the banks taking money from the citizens of Cyprus could cause widespread distrust amongst bank account holders all over Europe and the rest of the world. If the next European country having financial problems thinks that their bank accounts might be used as a last resort piggy bank; there could be a run on the banks all over that country. One thing that people all over the world share is a not very hidden distaste with anyone trying to toy with their money. A word to the wise. Find another way to fill the coffers. Or you might find yourself with a bigger problem to solve.
The European union is still such a young venture and the bonds that bind the countries together are very thin. There is no common government, no common language and a strong sense of individual identity. Right now they are choosing to remain together, but if one country leaves. It could provoke a chain reaction of countries leaving the euro and the union behind. If this happens, it could cause the euro to fall and cause a series of bank runs as everyone fights to not be the last one holding the bag. No one wants to have a bank account of worthless money. It could be terrible.
There will be a lot of talk about the need for unity - and with one failure, the whole system is in jeopardy. It's not true; the failure of Cyprus will tend to affect just Cyprus. It's a small country, and its problems are just a blip on the overall state of Europe.