The earliest known source is the Chicago Tribune columnist Mary Schmich, in a column of 1 June 1997 that later went round the world as “Wear Sunscreen”. Schmich has said she has seen no evidence that Eleanor Roosevelt ever said it. Roosevelt did write about facing fear, in different words.
What it means
Make a habit of small acts of nerve: each day, do something that makes you uneasy, so that fear stops deciding how big your life is.
Where it comes from
Schmich's column, headed “Advice, like youth, probably just wasted on the young”, opens “Inside every adult lurks a graduation speaker dying to get out” and goes on through a list of short pieces of advice, this line among them. By August 1997 it was being passed around without her name, retitled “Wear Sunscreen” and wrongly described as a graduation speech by Kurt Vonnegut at MIT. In 2012 she wrote that the line “is now widely attributed to Eleanor Roosevelt, though I have yet to see any evidence that she ever said it and I don't believe she did.”
Roosevelt did write about fear. In You Learn by Living (1960) she wrote: “You gain strength, courage, and confidence by every experience in which you really stop to look fear in the face … You must do the thing you think you cannot do.” Quote Investigator also traces an older family of sayings, from “Always do what you are afraid to do”, recorded in Emerson's essay “Heroism” (1841), to a 1961 novel by Mark Toby whose character is told to “do one thing every day that I want to do but I'm afraid to do.”