I think that backing a Nation's currency with physical and redeemable gold and silver broadcasts trust and faith that the currency is worth something and that makes it more stable in the overall marketplace, I think when Nation's are allowed to print without the backing of a physical commodity like Gold then it loses faith and value over time.
That is what they did to help get the economy back on track when the Great Depression happened. They told all the citizens that the government would buy everyone's gold jewelry . It worked so well they had to open up a military vault to store it all. It can be used in times when things really go down hill.
As gold has been used for a long time to be considered as value for trade, having such a thing in reserves would help balance out the economy is something were to happen. It shows there is money in reserve so people do not need to worry about a major crisis in most cases.
Yes, gold reserves make a nation's currency more stable, because when a currency is backed with gold, it prevents inflation. People and other nations can feel more confident doing business with the United States, if they know that their money is backed up. It makes the money that much more credible.
I do not think that a nation's gold reserves plays a part in making a country's currency more stable. I think that each factor is not really related to one another. I think that only think that will make a country's currency more stable is having a strong economy to go with it.