Make a gigantic safe made of Gold (or Silver) to store all the free Silver (or Gold) in. And it would be safe because it would be worthless and no one would want it.
Gold and silver becoming worthless would probably follow some form of horrible collapse of civilization. It's extreme, yes, but in that case people would likely not trade food or useful goods for metals that don't really have much use outside of an organized economy-based society. Of course no one really knows what would actually happen in such a situation, but I think many people overvalue the desirability of precious metals in an extreme collapse situation. In a mere economic collapse ("mere" only by comparison to a total collapse) precious metals might retain their value. But, again, as no one really knows what would happen the most we can probably say is "might retain their value." They also might not.
So civilization would have to devolve to the point where forms of payment for any type of good would have to be an item of immediate need, nothing that is transferable or a storable form of wealth. I think that we would have to devolve to cavemen (and cavewomen) again for that.
Silver could drop back to $5 if the economy booms. That is also the good news. If teh $20 trillion in debt eventually causes our monetary supply to becoem useless, then it could be the asset we all need it to be. Either way we win.
Probably half of the hoarders of precious metals , guns, and ammo would die within a year or two because they could no longer get their pharmaceuticals to keep them alive. Store the things you will need, not the things people tell you to accumulate.
Store the things you need that will keep. Many pharmaceuticals aren't especially durable; some need refrigeration, which would also be hard to come by. I remember a story by Niven and Pournelle (Lucifer's Hammer, I think) where one character was a sci-fi nerd who knew all sorts of things about how to rebuild technology -- but was diabetic. As I recall, he was trying very hard to help survivors rebuild to the point where he could get insulin, but it didn't work out.
I'd pick them out with my eyes closed, just by feel. Clad coins have a slimy feel to them compared to silver.
It only becomes worthless if a very large comet hits Earth and causes mass extinction. The Alien beings will finally risk landing and will plunder all silver and gold.
Doesn't have to be that way. Eventually with technology it may be possible to not only synthesize it but we could perhaps also develop tech to locate and find even trace amounts and then mine it with nano machines. Then there's the mining of asteroids or eventually other planets that could impact it heavily. I will agree though that people store things that aren't likely going to be as valuable as they believe in a collapse. I think food/water is the number one thing as the old 3/3/3, 3 minutes without oxygen, 3 days without water or 3 weeks without food and your dead comes into play rather quick once the grocery supply chain goes down.