In the first case, a crisis, we will see the dollar rally under flight to safety behavior. This behavior is what we saw last Fall with the strong dollar rally. The market wanted to avoid risk taking so money came to the dollar and dollar deleveraging increased.
At the other extreme will be a strong V-shaped recovery which will also see a strong dollar rally. Under this case, there will be stronger than expected growth, the US deficit will be closed, and the risks of quantitative easing will be diminished. While this may be a low probability event, it would be dollar positive. Money will flow to the dollar because of better investment opportunities.
The two most likely scenarios would be a tepid U-shaped recovery or a slow growth environment. Under the tepid case, there may be increased demand for risky assets in other countries. This may be a good emerging market environment. The other case o slow growth would be a normal recovery where growth fits within what has been expected by the consensus. who is thinking that we will be on a lower growth path. This certainly would not be a V-shaped recovery but a controlled growth. Under this scenario, there would be a desire to find risky assets outside of the US.
A problem with these scenarios is the underlying assumption that there is a decoupling between the US and the rest of the world. We have not seen this over the last few decades. The rest of the world will be able to move ahead without the US and the investors will looking to other parts of the world for the best returns opportunities. Given this world view, the dollar will diminish in importance. While we are not counting the dollar out, the dollar downside is greater if we have a economic muddle.