This is an important question. There are a few macro-level implications for a downgrade in credit. Before, I answer this question, let me say that many people agree with the credit downgrade. So, Paul Krugman is not the only voice. With that said, there are two main implications.
First, with the downgrade of credit rating, it will cost more money for the United States to borrow money. This will increase our debt and may worsen our debt problem in the longterm. This potentially worrying.
Second, the downgrade from the credit agency may send a message that the United States is not as strong as it once was in the area of being able to pay back its debt. This has an effect in the area of confidence. This may prohibit people and other nations from buying United States debt.