The American dollar is one of the strongest currencies in the world. More dollars will help avoid a financial crisis because more American firms are present in foreign markets and more Americans travel abroad with more American currency. Now that the Great Recession is over, dollarizing the world can commence.
Countries that move to adopt the dollar are not going to receive relief from financial hardship. It still boils down to what they can offer the global market in the form of goods. If they are in financial trouble all that excepting the dollar will do is give people more of a reference point into how bad their economy really is.