Presumably it would force them to lower it, which would be counter to the anti-inflationary moves of the Federal Reserve, but that might not matter given that this is a current issue. It's possible this might also affect banks willingness to raise rates in the future in response to Fed tightening if they thought there was a risk to the banking sector.
Given the size and how quickly the banks are failing I'd hazard a guess (this is not financial advice) that in order for the FDIC to maintain it's own portfolio it would have to raise rates enough to be noticeable to consumers, even given the number of FDIC accounts.
Can someone comment on if this is the case and how much this might affect forward guidance for banks and consumers?