From FAIR:

Responding to the 222-177 vote by the House of Representatives last night, adopting an amendment introduced by Representatives Michael Oxley (R-OH) and Barney Frank (D-MA), allowing banks and other financial institutions to accept the Mexican matricula consular document in lieu of valid U.S.-issued documents, FAIR issued the following statement:

"Once again, Congress has placed the priorities of a powerful special interest ahead of homeland security interests. In this case, the full House overrode the decision of the Appropriations Committee barring the acceptance of these documents and sided with the powerful banking lobby, which mounted an all-out campaign to have the ban against acceptance of the matricula card stripped from the final Treasury Appropriations measure.

"The House capitulated to the demands of the banking industry in spite of the fact that top law enforcement and homeland security officials have publicly stated that these documents are not secure and can be easily used by terrorists and criminal organizations to engage in money laundering. The desire of the banking industry to do business with people who are in the U.S. illegally has won out over the imperative to close the loopholes that have already been exploited by terrorists and criminal enterprises...

The voting breakdown is here. And, there's now overtaken-by-events backstory in "Texas lawmaker wants Mexico's ID cards barred from U.S. banks".

And, see "FBI Official Says Matricula Consular Card Is Security Threat".

And, for the view of the non-liberal press on these matters, see the example presented in the L.A. Times article "Barriers to Latino Home Buying". Despite its general title, it's partially about illegal aliens:

[A] study released Tuesday by the Congressional Hispanic Caucus Institute... favors the creation of a nationwide secondary market for mortgages sold to customers who do not have Social Security numbers...

...many undocumented Latinos believe they cannot open a bank account without a Social Security number.

In fact, in many states, including California, banks are allowed to accept the matricula consular, an identification card issued by the Mexican government through its consulates in the United States. Many banks also accept taxpayer ID numbers as a valid form of identification.