I have heard several accounts of why a low or falling dollar is bad:
1. U.S. citizens hold a relatively high percentage of dollar-denominated assets, so they are now poorer.
2. It looks bad when "the world’s strongest country" has a currency low in value. Perhaps OPEC will start pricing oil in terms of Euros.
3. Markets dislike uncertainty per se. People start wondering what a dollar is really worth and this causes them to hold off on other investments and purchases. This hurts financial markets and the economy more generally.
4. The real problem concerns interest rate hikes. The Fed won’t let the dollar fall too far, for some of the other reasons listed. It will stop a dollar free-fall by raising interest rates, which is bad for the economy.
5. If the dollar is falling, people will expect it to fall more and unload dollar-denominated assets. This one, however, is tricky.
If the dollar is expected to fall, we would expect nominal interest rates on dollar-denominated assets to rise (or the dollar must fall in value immediately). A reasonable equilibrium will obtain and dollars will once again be an attractive asset to hold.
My take: #1 is correct, but not a major problem. Imports are not a huge part of our economy, and often the exporter eats the currency loss, at least for a while. I don’t put much stock in #2. #3 and #4 are real. #5 makes little sense to me, but I cannot rule out its role in today’s world. How can it work? Perhaps portfolio managers bear a special penalty from being thought stupid if they hold onto dollars while a falling dollar makes the headlines. In this case a falling dollar would continually increase the real risk premia on dollar assets, even if traditional measures of risk do not much vary.
Keep in mind that the dollar did have a "soft landing" in the 1985-1989 period, so these are all possible costs, not necessary outcomes.
I cannot do links from this unusual Calcutta terminal, but read Brad DeLong’s recent posts on the dollar as well.