The FDIC insures bank deposits up to $100,000 per institution. So, if you have more than that, you might want to put it in different banks. Be aware that FDIC currently only has $50 billion to insure $1 trillion of accounts.
Brokerages generally have private insurance for accounts.
If we get runs on banks, and we could, then that money has to go somewhere. Such runs would probably happen to small- to mid-sized banks. While some of that money might find a hiding place under the mattress, it’s more probable that it will end up in a big bank.
When this debacle ends, and it will, a few giants will become even larger. Bank of America will almost certainly be one of them. Brokerages houses like Fidelity and Vanguard will also get bigger as the flight to safety accelerates.