Assalamu Alaykum wa Rahmatu Allahi wa Barakatuh
If there are Islamic banks that fulfill all the needs of Muslims in the community, many scholars would rightly argue that it becomes not permissible to deposit in a conventional Bank whether in saving or current account regardless of the issue of earning interest (earning interest is not permissible from conventional bank or from any other person). But whenever there are needs that cannot be fulfilled by Islamic banks, either because they do not exist or because their services are bad or insufficient, it is permissible to deposit in other banks provided a Muslim does not take or give interest which is exactly the Riba that is prohibited in the Qur’an.
One last point is worth mentioning: if you have to deposit large sums and to extended period, the Second Islamic Banking Conference held in Kuwait is 1982 determined that in such cases it is better to take the due interest and give it to charity than depositing in current account that does not generate interest or leaving it to the bank itself. This is based not on double standard but rather on weighing evils. That is when one is faced with two evils and has to choose one of them, the lesser evil is taking the interest and giving it to charity, not that we, without any need, deposit in a conventional bank and take the interest to charity.
Wa Allahu A’alam