Another SuperUser, @tagteam, pointed out to me that the coin and banknote collection is a special type of asset called a collectible. If you owned it for more than one year the profit is taxed at special rates, which might reduce the amount of tax you have to pay. You should report the sale as an investment sale. When TurboTax asks what type of investment you sold, select Collectible.
Not having a record of what you paid for the items is a problem, though. You have to know your "basis" (cost) in order to calculate the amount of taxable gain (profit). The fallback position when you don't know the basis of an asset is to use zero as the basis. But that treats the full selling price as taxable income, which would probably result in paying much more tax than you really owe. You should probably figure out a reasonable estimate of how much you paid, and keep a record of how you arrived at the estimate in case the IRS asks. If there's a significant amount of money involved you might want to get professional tax advice.
