Presumably a corrupt government imposes costs on people. There are literal financial costs, money lost to bribery and cronyism. And there are costs in freedom, justice, and so on.
Yes, you could say "the actions of corrupt governments" or "the policies of corrupt governments. But it's not necessary, and wouldn't really change the meaning of the sentence. The government is corrupt, and this corruption causes problems that the world must deal with.
As others have noted, it is more idiomatic to say "corrupt governments" than "corrupted governments". It is possible that the writer is making some distinction in context. "Corrupt" means that it is corrupt now, with no indication of past state. "Corrupted" implies that at one time it was not corrupt, but now it has become corrupt. This may or may not be a significant distinction, depending on the overall context.