The Bowery Boys have more on Roosevelt Island transportation options.
For most of its existence, people used ferries to get to and from Manhattan and Queens. Boatloads of prisoners, smallpox patients, the mentally insane, and, yes, residents of the island crossed the East River daily.Image is from The Bowery Boys.
Later, when the Queensboro Bridge sprung up on its north side, a trolley would stop in the middle of the bridge, allowing people to then enter a small elevator which would take them down to the island. According to NY Roads, this was the only way for the public to get to Roosevelt (then Welfare Island) in the 50s. I can't imagine this inconvenient from of commute brightened the island's reputation any.
A lift bridge spanning 2,877 feet to Queens was opened in 1955, finally allowing automobiles on the island. Its also the only way you can walk there. Those odd Queensboro elevators were dismantled in 1970.
However Roosevelt Island is often defined by its most popular method of conveyance, the Roosevelt Island Tramway. This unique way of getting to and from home, taking less than five minutes one way, is a picturesque and perfectly European way of experiencing the city. The aerial tram, made by the Swiss company Vonroll, is the only one of its kind on North America to be used as actual mass transit. (Many vacation destinations obviously use trams, including mountains in Oregon and New Mexico.)
UPDATE - 12/10/08 - Here are directions to Roosevelt Island.