Jasamine Vieira lives in Brownsville, Brooklyn, but everything she needs is everywhere else.

Ms. Vieira, a nurse, sends her 7-year-old son to school in Crown Heights instead of the troubled schools in her neighborhood, and to playgrounds in Williamsburg, away from the shootings she often hears about. Standing in the spice aisle of an Associated Supermarkets in Brownsville on Sunday, she said she preferred to trek to wealthier neighborhoods for the organic food and better quality meats she wanted to feed her son.

On Thursday, Gov. Andrew M. Cuomo announced a $1.4 billion plan that would bring much-needed resources, such as health care services and new jobs, to Brownsville and other long-suffering areas of central Brooklyn, which includes East New York, Bedford-Stuyvesant and Crown Heights. The plan would pour $700 million into health care in those neighborhoods where obesity and asthma rates eclipse state and city averages. Mr. Cuomo said the plan also would create 7,600 jobs in the area where unemployment is high.

Residents in central Brooklyn met the announcement with a mix of elation and skepticism. Parents who keep their children mostly inside for safety warily embraced it. Others worried about the potential for gentrification, leading to the displacement of the area’s mostly black and Latino residents.