ANSWER
Seven:
Kelsey Grammer
Cheers: Frasier Crane
ST: Capt. Morgan Bateson (ST:TNG episode "Cause and Effect")
Kirstie Alley
Cheers: Rebecca Howe
ST: Lt. Saavik (STII:The Wrath of Kahn)
Bebe Neuwirth
Cheers: Lilith Sternin-Crane
ST: Nurse Lanel, an alien (ST:TNG episode "First Contact")
Kate Mulgrew
Cheers: Councilwoman Janet Eldrige, dated Sam (episodes "Strange Bedfellows: Parts 1, 2, and 3")
ST: Capt. Kathryn Janeway
Brent Spiner
Cheers: Defendant Bill Grand, nearly convicted by Diane (episode "Never Love A Goalie: Part 2")
ST: Lt. Cmdr. Data
Christopher Lloyd
Cheers: Artist Phillip Semenko, painted Diane (episodes "I'll Be Seeing You: Parts 1 and 2")
ST: Klingon Cmdr. Kruge (STIII:The Search for Spock)
Mark Harelik
Cheers: Reed Manchester, Diane's friend (episode "One for the Road")
ST: Inspector Kashyk (ST:Voyager episode "Counterpoint")
With thanks to "Jennifer Harper's Cheers Page" (http://cheers.tvheaven.com/) for this compilation, with additional information from the Internet Movie Database (http://www.imdb.com).
