Seeking a shot in the arm for the ailing banking business, Citigroup Inc. is expanding into a little-known but fast-growing field known as identity proofing—the tedious and time-consuming task of proving people are who they say they are.

The third-biggest U.S. bank by assets later this month will begin issuing digital-identity badges to the employees of Defense Department contractors, ranging from makers of high-tech engineering parts to the janitors who clean the bathrooms. Citigroup is the only financial institution that...