The hospital where Kate takes Claire when she goes into labor (“What Kate Does,” Ep. 6×03) is Hawaii Medical Center West, formerly St. Francis Hospital, at 91-2135 Fort Weaver Road. The street number on the building is briefly visible in the background as the camera follows Kate from the taxi cab. The interior scenes (featuring Kate, Claire, and Ethan) were also filmed here.
Note that the hospital later used for most of Season 6 was the Rehabilition Hospital of the Pacific in Nuuanu.
Map Coordinates: 21.371744, -158.02721
Image copyright ABC Studios. Courtesy Lost-Media.com.