I worked for almost a year for Provident Bank of Maryland (at several Virginia branches), and we tellers were required to accept the Mexican Matricula Consular as a valid form of ID for check cashing.
Before New Year, I helped out at a Richmond branch, and we had some questionable folks coming in to cash payroll checks. One presented to me his check and a New York Driver's License that did not feel legitimate, so I asked for another form of ID. The man scowled at me and pulled out a Matricula Consular.
The business that my dad used to work for (he was the office manager) hired plenty of illegals. They banked with SunTrust Bank, which was doing brisk business cashing payroll checks.
Believe me, the banks (and now credit unions) do NOT care about your legal status so long as they can make money off you (without the Feds going after them).
Thank goodness I now work for a large military credit union (in their IT department). We don't have that many illegals coming to our branches :P