Any profit on the sale of a personal item is taxable income, and you do have to report it. You might have a profit on the coin and banknote collection. Note that the taxable income is just the profit, not the full selling price.

You probably have losses on the books because, as in the example you gave, you are selling them for less than you paid for them. You cannot deduct a loss on the sale of personal items, and you do not report the loss on your tax return.

Since this is a one-time activity you can report any profits as "other income." You do not have to report it as a business.

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.

@Anonymous, are you a citizen of Canada, filing a CA tax return?

This is the US forum....

Here is the Canada website
http://turbotax.intuit.ca/tax-software/index.jsp 

TT Canada Forum
https://turbotax.community.intuit.ca/tax-help

TurboTax Canada Support 

https://support.turbotax.intuit.ca/contact/

I'm going to move your question to the Canadian Community, where you would be better served. The answers that you've received so far are all correct concerning US tax law, but we wouldn't know how you should proceed.