Do what you feel in your heart to be right — for you’ll be criticized, anyway. You’ll be “damned if you do, and damned if you don’t.”

Eleanor Roosevelt (1884–1962) First Lady of the US (1933–1945), politician, diplomat, activist

Quoted in Dale Carnegie, How to Stop Worrying and Start Living, ch. 21 (1948)
    (Source)

I cannot find this in any of Roosevelt's written works or other primary sources. Carnegie wrote that Roosevelt told him this personally, and framed it as an extension of advice she'd received from Anna Roosevelt, Theodore Roosevelt's sister: "Never be bothered by what people say, as long as you know in your heart you are right."

From this Roosevelt has sometimes been credited with originating the phrase "Damned if you do and damned if you don't," though she set it in quotation marks to indicate she was quoting it. The saying was actually first written by evangelist Lorenzo Dow in 1836.