Comments
I join the chorus of those who are confused by the ambiguity of this question. On the other hand, considering the role of banks and financial systems in our economy and social lives, I'm surprised, too, that we rarely mention them in EA.
If you're thinking about the impact of microcredit, unfortunately, there's no evidence that it's very effective - though microinsurance might be a low hanging fruit. You might also be interested in how banks prey on people who lack the skills and knowledge necessary to make financial decisions; however, there are some initiatives about this - I remember Marginal Revolution and Future Perfect citing how economic undergrads were learning about financial education, and some government agencies are often concerned about it, too. Moreover, it's an interesting way to present some concepts about rationality, like risk-aversion, cognitive bias, time preference, etc. Besides, governments and regulators are often concerned about predatory practices, concentration, stability and efficiency, and the whole market will supposedly change with technological innovations. I'd like to see some evidence on the effectiveness of their policies (too many people talking about "financial disruption"), but the area is far from being neglected.
If you're thinking about systemic change (and the systemic risks our financial system entails), I strongly recommend The End of Banking. I wouldn't say this subject is like an x-risk, but economic crises are frequent hazards that greatly affect economic growth (and probably politics, too). Open Philanthropy dedicates some attention to macroeconomic policy research, and there's a bunch of smart people everywhere concerned about it. Again, not neglected.
Or maybe you're thinking about how one can use banks to gain leverage and amplify the impact of some policy; for example, since they're often seen as the bad guys, many banks (and related associations) are announcing committments concerning climate change mitigation and corporate social responsibility, and people are discussing how climate hazards should affect their balance sheet. This approach is probably limited, though; but maybe there's a low-hanging fruit in nudging those ethical washing activities into effective charities.
Finally, another attractive feature of financial systems: they provide interesting examples of success and failures in governance. Central Banks are monetary and regulatory authorities, whose power relies on expertise and financial tools, instead of popular support or political representation; and a bunch of international financial organizations, such as IMF and BIS, are often more successful in implementing policies than institutions receiving more attention from diplomats and heads of state, such as UN and OAS - think about how the EEUU depends on the ECB, for example. I wonder if this could offer a case study for AI governance (or other x-risks).
I suspect the choice of bank is rather unimpactful, even for those wtih a few million dollars in deposits. For most of us, it's really not worth time trying to optimize - you're better off finding a site that reviews banks and compares fees, etc. But if you are concerned about the systemic risks and externalities imposed by banks, I would recommend finding a credit union rather than a bank - or at least find a small commercial bank rather than a large national bank or an investment house for banking. (But again, I suspect convenience and fees are a more important factor.)
Edit: To clarify a bit, the marginal impact of giving money to charities is significant, while the marginal impact of giving your savings to a bank is fairly minor - it just gives them a slightly larger balance sheet to make loans, though most are not exactly short on cash nowadays. But if you want to think about systemic change for banks as potentially an important issue, picking where to put your money isn't as important as contacting your senators to tell them you want them to regulate banks for tightly.