You needn't look any further than many failed states. When a country can't pay its debts, the typical response of the government is to just print money. This causes Hyperinflation and causes everyone to try to divest themselves of the useless currency. In other words, the currency itself becomes worthless.
Assets still have the value they always had, such as food, clothing, and fuel. Some items that have real value but are also durable and portable may actually rise in value because they can be used as a new form of money. Historically this has been gold, but it could be anything. Cigarettes are a form of currency in prisons. Batteries, non-perishable food, and medicine seem like great candidates.
A lot of people's investments aren't actually money, but stocks. If you own a stock, you own a portion of a company, which is really an asset. Assuming that company can figure out how to stay in business and produce their product for some kind of profit, then that stock still has value. 207
If nothing else, that stock may entitle you to a share of their assets if they liquidate (but it depends on the type of share).
There are other alternatives to cash though. Let's say as soon as the aliens started dropping money, the government declared that all cash was worthless, but all deposits were still valid. A bank, for instance, only keeps 5% of the on-deposit money as cash. When you deposit \$100 in your account, the bank loans out \$95 of that, and that person typically deposits it in their bank account (or spends it meaning the seller deposits the money) and then the bank loans out 95% of that \$95, and so on. The net result is called the "money multiplier" and it ultimately means that the amount of money in circulation is only 5% cash. It's hard for me to understand what this would mean, if suddenly only that 5% became worthless. Could we just switch to a cashless society? How does the government continue to manage the money supply?
One option would be something like Bitcoin. One could imagine a crypto-currency where only the government had the keys necessary to authenticate a transaction. This would be some kind of centralized electronic currency (instead of the distributed nature of Bitcoin).
Countries have gone through periods of Hyperinflation and then stabilized by issuing a new currency. If you immediately say "cash" is worthless but money on deposit has value, and you immediately start issuing a new digitally-authenticated currency where the value was one-to-one the same as your existing denominations, and you offered some kind of replacement plan for the banks that hold most of the cash currency (the amount they have on hand at the moment the aliens started dropping cash could probably be authenticated by an audit) then it just might work, and the world could avoid big financial system problems. 852