Meacham appeared on Broadway in The Long Watch for which she received a Clarence Derwent Award, Ondine, Eugenia, The Legend of Lizzie, Moonbirds, A Passage to India, The Seagull, The Crucible and Rosencrantz and Guildenstern Are Dead. While she never appeared on Broadway in the works of Tennessee Williams, she may be best known in the theater world for her association with the playwright. She won an Obie Award for her performance in his Suddenly, Last Summer and also appeared in Williams' In the Bar of a Tokyo Hotel and The Gnadiges Fraulein. Meacham also won an Obie for her performance in a 1961 staging of Hedda Gabler. Among other notable credits, Meacham spent a long time on the soap opera Another World, in which she played eccentric maid Louise Goddard.