I doubt it, especially given where gold is on the Periodic Table. You might have locally abundant concentrations of gold or silver, Potosi comes to mind, but overall, I do not view it as a problem. I am not sure that asteroid mining for gold or silver is going to be a big thing, but silver as a by-product is a possibility.
Some of the copper mines in the Upper Peninsula of Michigan had considerable quantities of silver as a natural alloy with the copper, but the smelter did have to separate them out. Overall, the influx of possibly cheap gold or silver would simply serve to increase the money supply, possibly causing inflation in some areas, but you are talking about multiple planets here. That should serve to localize the effects.
I am also not sure if asteroid mining is going to be such a cheap source of the rarer materials. Nickel-iron asteroids are likely common, but you still have to separate the nickel from the iron. The rarer elements are more likely to be found in the stony asteroids, and you have to get the ore from the asteroid to a smelter. That takes energy, and energy in an asteroid belt is not going to be cheap. Plus, the smelter you want might be on the other side of the belt, several hundred million miles away.