Signature announced on Friday that it was stepping up efforts to make sure its lending doesn’t lead to the displacement of tenants. The pledge drew praise from the Association for Neighborhood & Housing Development, a New York community group.

Last week, Gov. Andrew M. Cuomo of New York said the state was opening an investigation into allegations, filed in a lawsuit, that Kushner Companies illegally harassed low-rent tenants to get them to leave.

Christine Taylor, a Kushner Companies spokeswoman, said, “All our business with Signature Bank has been entirely appropriate.” She added that before Mr. Kushner joined the White House, “we never had these type of inquiries which appear to be solely for political reasons.”

Amanda Miller, a Trump Organization spokeswoman, played down the company’s relationship with Signature. “While the company has, from time to time, done some business with Signature Bank, those dealings were few and far between and limited in scope,” she said. (Other banks, such as Deutsche Bank, have done more business with the Trump Organization.)

Mr. Shay and Joseph DePaolo founded Signature in 1999 with backing from Israel’s biggest lender, Bank Hapoalim. One of Signature’s specialties was financing the purchase of taxi medallions, which authorize holders to operate cabs. Mr. Cohen had amassed a large portfolio of medallions and had borrowed money from bankers who later joined Signature.

Signature forged deep political connections — its board members have included a former Republican senator, Alfonse D’Amato, and a former New York lieutenant governor, Alfred B. DelBello, a Democrat. A former Democratic congressman, Barney Frank, joined the board in 2015.