He was a leading advocate of lowering the U.S. voting age and wrote the [[26th Amendment]].<ref name=nytimes/>
Randolph faced a strong formidable challenge in his 1978 re-election campaign from Governor Arch A. Moore.<ref>Franklin, Ben A. (May 28, 1978). [https://www.nytimes.com/1978/05/28/archives/toughest-race-in-decades-for-west-virginia-senator-an-irreverent.html Toughest Race in Decades for West Virginia Senator]. ''The New York Times''. Retrieved August 2, 2021.</ref> Although he heavily outspent Moore,<ref>Roberts, Sam (January 8, 2015). [https://www.nytimes.com/2015/01/09/us/arch-moore-91-w-virginia-trail-blazer-dies.html Arch Moore, Trailblazing West Virginia Governor, Dies at 91]. ''The New York Times''. Retrieved August 2, 2021.</ref> Although he heavily outspent Moore, Randolph only won the general election by one percentage point.<ref>[https://www.ourcampaigns.com/RaceDetail.html?RaceID=5331 WV US Senate Race - Nov 07, 1978]. ''Our Campaigns''. Retrieved August 2, 2021.</ref>
Randolph gave support in 1982 to a proposed gasoline tax as a means to fund a transit program.<ref>Tolchin, Martin (December 8, 1982). [https://www.nytimes.com/1982/12/08/us/senate-democrats-urge-works-bill-as-a-substitute-for-road-job-plan.html SENATE DEMOCRATS URGE WORKS BILL AS A SUBSTITUTE FOR ROAD-JOB PLAN]. ''The New York Times''. Retrieved August 2, 2021.</ref>