Whether we can keep cash we have found and what we can do with it are ethical as well as legal questions. Even though cash is not marked with the owner's identity—like a check or savings bond—it is a piece of property that originally belonged to someone other than the finder. So technically, cash you find is not automatically yours.
Many communities have local laws or ordinances governing what a person must do if he finds cash and does not know who it belongs to. These laws usually require that a person who finds money, especially larger amounts (for example $100 or more), turn it over to the local police. If no one claims it after a certain period of time, the police can then give it to the finder to keep. Some communities may have different laws and some have none. In the US, traditional ethical guidelines about finding money are quite similar—you should try to find the owner and only keep the money if you made a reasonable effort and couldn't locate the owner.
If you find money, especially a significant amount, you should check your local laws or contact an attorney or the police. If a law requires that you turn over money you have found to the police and you do not do so, you could be charged with larceny or theft. Even though you did not steal the money by taking it directly from its owner, you are holding the money and not trying to return it. Holding or possessing property that you know does not belong to you also constitutes theft or larceny under most state laws.
You also can make your own efforts to identify or locate the owner of money you have found. If you find a $20 bill on the sidewalk, it is common courtesy to ask others walking nearby if they might have dropped the bill. There is always the risk that someone will say they dropped the money just to be able to claim it. In those kinds of situations, you have to use your own judgment about whether the person claiming the money is being truthful.
If the money is in a wallet, you should check for ID. If the wallet contains no ID but does have credit cards, you can call the customer service number on a credit card and the card company can contact the owner. You also should look inside a container of cash for any information it might contain about the owner. With regard to the money found behind the basement wall, you could try to contact the previous owners of the home and ask if they knew anything about any property being stored behind the basement walls.
Especially if you find a large amount of money, it probably is best to contact an attorney who can verify local laws regarding lost and found cash and advise you on how to proceed.