Well, he did, in October 1958! It was an election year, and Harriman, a Democrat, was running for his second term as governor against Nelson Rockefeller, Republican. Harriman was an interesting man. His father was a wealthy railroad baron, and when Harriman was a boy he went on his father's "Harriman Alaska Expedition," traveling on his father's yacht with a party that included such figures as John Muir and Edward Curtis. In the WWII years he was our ambassador to the Soviet Union.
However, unfortunately for Harriman, but perhaps fortunately for SUNY, Rockefeller won. Over the next decade or so he presided over a massive expansion of the state's public higher education system that saw schools like ours grow at a phenomenal rate. Brockport for example went from 1000 or so students in 1958 to over 10,000 in the late '60s!