We don't have to indulge in wild speculation. While there haven't been any crises that wiped out the entire world since the Flood, there have been plenty of lesser crises that wiped out a nation, like invading armies or particularly large natural disasters.
And what happens? There's no reason for money to still have value, as the government that backs it is going down. But in practice it doesn't become worthless, but prices skyrocket. Some number of people will still accept money in exchange for something that will actually help you escape, but they demand huge amounts of money. I guess this makes sense if you assume that the probability that the nation will be destroyed is not 100%. If you think there's some small chance that the nation will survive and money will still have value, then it makes sense to still accept money, but the smaller you think the chance of the government surviving, the less the money would be worth.
Rich people have not just cash in the bank but things: cars, planes, or boats that can be used to escape, large tracts of land to hide, shelters, etc. Sure, a mob could attack and steal these things, but the rich person has the advantage that he's holding them now. For someone to steal it, they have to know that he has, and they have to know where it is, and they have to overcome whatever effort he makes to defend it.
If I was rich and I saw a global disaster coming, I'd start immediately on building a shelter or escape rocket or whatever. As long as the disaster is far enough away that people still have lives to lead in the meantime, money will still be of value to them. And as long as there is some question about whether the disaster will really happen, people who doubt it will gladly accept money to help build your shelter. After all, there have been many predictions of apocalypse over the centuries, and to date the world is still here, they've all proven false. So depending on the nature of the disaster, there may well be plenty of people who will doubt that it will really happen. In the movies, of course the disaster really does come, and the doubters are all shown up as fools for failing to listen to the warnings while they had the chance. But in real life, usually the disaster does NOT come, and the people who sold everything they owned and fled to a shelter in the wilderness are laughed at. As long as there's doubt, money will still have value.
Oh, rich people often have one other asset: leadership skills. When everyone is in a panic, someone can often walk in and take charge. If you can put on an air of authority, act confidently, and have the competence to do it, people will listen to you. Lots of people throughout history have managed to convince armies of people -- and literal armies -- to follow them using the shear force of their personality.