This is especially apparent on the Lower East Side, which has a high number of banks compared with some parts of the Bronx and Queens. When the Lower East Side People’s Federal Credit Union received its charter to open in 1986, it replaced the only bank branch within a 100-block area. Now, Pew data show that while the neighborhood has one of the borough’s highest concentration of non-banking households, nearly every major bank has opened a branch there.

Image At left, Humberto Vargas, a restaurant owner, has been trying to persuade one of his deliverymen, Che Che Matos, right, to join a bank and build a credit history. Credit... Michelle V. Agins/The New York Times

Dozens of groups and neighborhood bankers know firsthand how hard it can be to get New Yorkers to trust banks. The New York Urban League, which has worked for five years on a program it calls “financial connections,” found that it had trouble drawing participants when it was marketed as banking education. Repositioned it as a money-management training program, it had more of an eager audience.

Workers at Amalgamated Bank, which opened a branch in an underserved section of Long Island City two years ago, continue to answer questions from potential customers who came from countries where friends and relatives lost their fortunes to collapsed or corrupt banks.

“There isn’t that trust that Americans are accustomed to having,” said Peter Mosbacher, the bank’s senior vice president of community development. “We are having that challenge to get people to understand that the American banking system is stable.”

Nearly two years ago, the city’s Department of Consumer Affairs started offering accounts without overdraft fees that have drawn 1,700 low-income families into the banking system, the department said. An additional 1,207 New Yorkers had signed up for the agency’s $aveNYC Account Program, a program that provides savings accounts that is active only during the tax season.

With a big hurdle still ahead, some politicians, government agencies and banks are turning to Bank On, a program that has been adopted in 60 cities nationwide. The Manhattan borough president, Scott M. Stringer, organized a forum for 71 executives from large and small banks, community leaders and politicians at the Federal Reserve last month to begin the program in New York City. While Mr. Stringer agreed that the task will be especially difficult, with financial institutions having lost so much credibility in recent months, he argued that it may be easier to persuade hard-pressed banks looking for customers to reach out to these New Yorkers now.

“Sometimes when times are most difficult, banks realize they have to expand their customers,” he said.