When historic flooding ravaged southeastern South Dakota last summer, leaving homes and businesses in ruins, Gov. Kristi Noem made a controversial decision: She chose not to deploy the state’s National Guard.

Citing the high cost and arguing that the Guard should only be called for “a true crisis,” Ms. Noem left thousands of residents to cope with the aftermath without the additional support, despite widespread devastation and mounting calls for help.

“We have to be wise with how we use our soldiers,” she said during a June news conference in Yankton, S.D.

Her reasoning shocked residents and lawmakers across the political spectrum, particularly in light of Ms. Noem’s earlier decisions to spend millions of taxpayer dollars to deploy the Guard to the U.S.-Mexico border to address the immigration crisis.

That move was one reason Ms. Noem was picked by President-elect Donald J. Trump to lead the Department of Homeland Security, a sprawling department that includes Customs and Border Protection and various other immigration-related agencies.