The campaign begins today - exactly 46 years since Armstrong’s moonwalk in 1969.
Experts say spacesuits were built for short-term use with materials that break down over time.
The museum aims to raise 500,000 US dollars (£321,000) on Kickstarter to conserve the spacesuit, build a climate-controlled display case and digitise the suit with 3D scanning.
The Smithsonian has formed a partnership with Kickstarter for a series of crowd-funded projects, with the spacesuit being the first.
Armstrong’s spacesuit is deteriorating and has not been displayed since 2006.
The museum plans to display it for the 50th anniversary of Armstrong’s moonwalk. It will then become the centrepiece of a “Destination Moon” exhibition due to open in 2020.