Doesn't matter. All depositors protected up to $100k (investment accounts are usually insured for much more) by the FDIC...as long as you think the US is solvent (questionable, at this point, but we're all working with USD, so doesn't matter where the dollars are...if they become worthless they'll be worthless no matter what).
Choose a bank that's got a branch near you...you'll be there more often than in the past. National, or at least in the major cities you can imagine finding yourself in. We've got an account with WaMu, who just merged into JP Morgen (which kinda sucks, as banks go) due to insolvency. I don't plan to move to another bank until the policies and fees change to the JP Morgan Chase standard (which they will), and folks start showing up in the uniforms and trying to sell me something every time I walk in the door.