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 ʻalayhi wa sallam, is His slave and Messenger.
In brief, saving money in usurious banks is considered an interest-based loan. The client loans his money to the bank to receive it back with an agreed-upon interest. The usurious bank invests this money in prohibited transactions by giving out loans at higher interests, thus making profit and giving part of them to the client as interest. This is unlawful, and it is impermissible for the Muslim to partake in it. As for Islamic banks, the savings account in them is based on a mudhaarabah contract between the bank and the client whereby the bank invests the money in lawful activities and gives the client a percentage of the profit, such as one third or one fourth, without guaranteeing the capital if there is loss, provided that there was no negligence or transgression on part of the bank. The mudhaarabah transaction was explained and detailed in fatwa 201160.
As for the client's ignorance of the activities in which the usurious bank invests his money, it is irrelevant as far as the prohibition in this regard is concerned. It is impermissible for the Muslim to partake in a contract with the interest-based bank to begin with, even if the bank invests the funds in lawful activities or does not invest them at all. This is because the legal implications of the contract entail that the bank guarantees the invested funds and pays the client a fixed interest rate. This is plain interest that has been deemed prohibited by the sharee'ah. The approval of the contracting parties of it does not waive the prohibition of such a transaction.
Allaah, The Exalted, says (what means): {O you who have believed, fear Allaah and give up what remains (due to you) of interest, if you should be believers.} [Quran 2:278]
There are many sources of lawful earning for whoever keenly seeks after them. When a Muslim needs to deposit his money in a usurious bank given the lack of Islamic banks, as is the case in most non-Muslim countries, then he is allowed to deposit his money in current accounts only in order to avoid usury, and he should not opt for that except if there is a need for it.
Allaah knows best.