First some history. The US did not always have a monetary policy. It has not always had a strong currency. It defaulted in the Revolutionary War and in the War of 1812 and on the Bretton Woods gold standard agreement.
With the enactment of the National Banking Act of 1863, during the American Civil War and its later versions that taxed states' bonds and currency out of existence, the dollar became the sole currency of the United States and remains so today. (I've highlighted Civil War here because the US system is coming from a war, not from a consensual union. Politically speaking this has consequences for the EU to consider if it wants to be more like the American model)
The US did not have a proper "monetary policy" until the Federal Reserve was established in 1913, the last of all industrialized nations at that time.
Debt is the main problem facing the EU.
After the Civil War, the Fourteen Amendment, said that US debt (Green Backs) can not be repudiated. It must be paid in other words. It also said that debts incurred by the Southern states from the war or loss of slaves cannot be paid. Ten states defaulted. Nine states defaulted during a major recession in the 1840s. The Great Depression caused a new round of state bankruptcies. The effect of this is that the US states have nearly all laws that require them to have balanced budgets and it is well followed. This has occurred as a bitter lesson of history, so these things are not always transferable.
One intent of the EU's Maastricht Treaty was to keep down debt levels of member states, but it was not carefully followed. US states are not likely to keep hardly any debt. The US states aren't nations either, so they do not have external debt usually. The US constitution does not allow US states to sign treaties with foreign countries without federal approval, for instance. High EU member debt levels can bring down the Euro.
US redistributes money to keep up its monetary union.
EU members may have started to notice that some nations have been asked to pay for other countries over spending.
In the 20th century, US states have no longer operated their own armies or welfare systems or they are heavily subsidized. Federal tax monies are constantly redistributed from wealthy to poorer states through various mechanisms. Economists theorized that the redistribution of workers crossing borders in the EU would be sufficient to accomplish this goal. Fewer workers moved than was believed would. Europe speaks many languages and has many cultures and has many blocks to entry to jobs in other countries even if now one may freely move from one nation to another. Living standards did not improve in less wealthy countries through this mechanism and it failed.