OK, here's my understanding of all of this (could be wrong...)
The dollar has been the dominant currency in the world for one main reason - America has been the most dominant trading nation.
As for dollar stability, America has done much to undermine the stability of their own currency. The budget deficit and current account deficit means that as soon as financial flows reverse out of America, the dollar will take a hit (as it has recently done). Anyone using dollars is of course affected by such changes, even though they had nothing to do with them. This can make the dollar less attractive as a trading currency: if it is volatile, people will avoid holding dollars for too long.
After World War II and the Bretton Woods conference, the world financial system used the gold standard which was pegged to the dollar. This system gave the Americans a lot of power over the world economy. They caused a lot of damage by effectively printing too much money, causing the agreed dollar-gold exchange rate to become unrealistic, and the dollar to be effectively overvalued. When the system was discontinued, the dollar lost about half it's value (as it had been overvalued for some time). The Latin American economies had pegged their currencies to the dollar and so they also lost half their currency's values. America had effectively exported its inflationary financing to other countries!!!
This is the problem with having a universal currency - the first thing you need to ask is who is going to be in charge of the money supply and controlling inflation, as it can otherwise be abused. A currency's value is determined by these factors.
If America's financial woes continue and American trade declines, there will certainly be a decline in demand for the dollar - but never count the Americans out. Hopefully they get their act together very soon!