Before the days of the large international bank, most people had their financial needs met at the corner savings and loan. If big banking has lost its appeal, seek out smaller, local banks, many of which aren't publicly traded, or credit unions, which are not-for-profit. The difference between these two types of banks and large, publicly traded ones is that banks that don't have to appease shareholders can focus on its customers first.
According to the Independent Community Bankers of America, local banks focus on "personal service, local credit decisions and ownership, and reinvestment in the community." And according to the Credit Union National Association, credit unions exist to provide financial literacy for their members, serve the needs of their members regardless of means, and offer lower rates than traditional large banking models.