Stone began his acting career in 1976 as a child actor on Charlie's Angels. However, most of his acting roles were as an adult. He appeared in two episodes of Space: Above and Beyond, and did two film roles. His most notable performance, however, was as the hapless gay Los Angeles millionaire Michael Beebe in the second-season episode Beware of the Dog on the television series Millennium.

However, Stone's primary career was as a casting Director. He began this career with The Landsberg Company in 1981. His first job was casting the NBC series Gimme A Break!. He was head of casting at 20th Century Fox Television, and was responsible for casting David Duchovny and Gillian Anderson on The X-Files and Lance Henriksen on Millennium. In 1998, Stone, who was gay, was terminated at Fox and filed a complaint with the California State Labor Commission that he had been illegally discriminated against on the basis of his sexual orientation. The company refused to comply with the Commission's decision.

He was nominated three times for an Artios Award by the Casting Society of America. In 1982, he was nominated for Best Casting for Comedy Episodic TV for his work on Cheers. In 1986, he and co-casting Director Lori Openden were nominated in the same category for casting the pilot episode of All Is Forgiven. And in 1991 he was nominated for Best Casting for a TV Miniseries for Switched at Birth.

Among his more notable film and television casting jobs were the film Jaws 3-D (1983), the made-for-TV movie The Ryan White Story (1989), Cameron Crowe's directorial film debut Say Anything... (1989), the made-for-TV movie The Incident (1990) and the television series Space: Above and Beyond.

In addition to his Oscar, he and fellow casting Director Holly Powell won an Emmy Award in 1990 for Outstanding Achievement in Casting for a Miniseries or Special for The Incident. Stone and Powell were only the second recipients of this award, which had been established in 1989.

In 1998, Stone co-founded a nonprofit organization inspired by the film Trevor, called The Trevor Project. The organization runs a 24-hour, toll-free suicide prevention hotline aimed at gay and questioning youth in the United States. The organization produced teaching guides and support materials for distribution to teens in schools.

In later years, he also produced several films. He was executive Producer for the film Little Man Tate, Jodie Foster's directorial debut motion picture. (During the 2007 Academy Awards, Foster referred to his death, and called him her "best friend.") He and co-producer/director Peggy Rajski won an Oscar for the 1994 short film Trevor, a comedy-drama about a gay teenage boy's attempted suicide.Ellen DeGeneres hosted a special airing of the film on HBO in 1998. In 2006, Stone wrote and executive produced the made-for-TV movie A Little Thing Called Murder, starring Judy Davis, based on the story of murderer Sante Kimes. It won him the International Press Academy's Satellite Award for Motion Picture Made for Television.

The 2008 motion picture, The X-Files: I Want To Believe was dedicated in his memory in the closing credits.