It is not permissible to put money in riba-based banks except when there is a need for that and when there are no Islamic banks. In that case one should only use current accounts that bear no interest, as a case of committing the lesser of two evils. Otherwise opening a current account in a riba-based bank is also haraam, but the prohibition is less severe than in the case of a savings account. The reason why it is haraam is that the bank will make use of this money and use it to help in its haraam transactions such as riba etc.
But if a person fears for his money, then it is permissible for him to put money in these banks.
It says in Fataawa al-Lajnah al-Daa’imah (13/346):
It is not permissible to put cash etc in riba-based banks and other riba-based institutions, whether that is with or without interest, because that is a kind of cooperating in sin and transgression, and Allaah says (interpretation of the meaning): “do not help one another in sin and transgression” [al-Maa'idah 5:2]. But if he fears that his wealth may be lost, due to stealing, robbery and the like, and he has no way to protect it other than putting it in a riba-based bank, for example, then he is granted a concession allowing him to put it in the bank or other riba-based institution, without any interest, so as to preserve it, because that is committing the lesser of two evils. End quote.
If putting the money in one account will pose a danger to it, then there is no reason why you shouldn’t open another account – without interest – after making sure of what the bank is claiming. If the danger can only be removed by opening a savings account, that is permissible, so long as one does not benefit from the interest and donates it to Muslim charities instead, such as schools, hospitals, helping the poor and needy, and so on.
But you should not do this until you have made sure that this danger is in fact real, as stated above, and made extra sure of that, because of the severity of the prohibition on lending with riba, because the Prophet (peace and blessings of Allaah be upon him) cursed the one who consumes riba and the one who pays it, and it is well known that savings accounts are regarded as a loan from you to the bank, and this is a riba-based loan.
And Allaah knows best.