Might I suggest reading this Investopedia article on negative interest rates?
Negative interest rates fall into a couple different categories. If a central bank has a negative interest rate policy in place, it is their way of "encouraging" lending, since a negative interest rate means the banks have to pay for un-loaned capital they hold on their books. It's a sort of penalty for not being more liberal with lending policies.
Rarely will you see negative interest rates being offered by banks to customers, especially in the U.S.
Banks in the U.S. are constrained in their lending by their capital reserves, which are dictated by customer deposits. So, a bank can only loan a certain amount of money (usually a multiple of its capital reserves) and is dependent upon more deposits by its customers to add to that amount. Charging customers to keep money in the bank would have the opposite effect, chasing customers from that bank to competitors who either don't charge a negative rate at all or whose negative rate (assuming everyone is charging it) is less.
I don't see such a thing happening here in the U.S., if for no other reason than it is bad public relations. Better to offer no interest at all than to tell customers they have to pay to keep their money in its accounts.