All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is not permissible for a person to deposit his money in a Riba-based bank because this consists in cooperating with the bank in the sin that it is committing even if one does not receive any interest in return for depositing his money. Depositing money in the account (in such banks) helps the bank because the bank benefits from the money that is deposited in it. Indeed, it is not permissible to help a sinner in sin. Allaah says (what means): {And cooperate in righteousness and piety, but do not cooperate in sin and aggression.} [Quran 5:2]
However, if a person is in a country where there is no Islamic bank, and he fears losing his money, then it is permissible for him to deposit his money in a Riba-based bank, but in a current account, as this is less serious than other types of accounts.
The services offered by the bank in exchange for depositing money in it for a certain duration consists in Riba. This is because the owner of the money lends his money to the bank so that he benefits from those services. Indeed, any loan that causes a benefit consists in Riba.
Allaah Knows best.