Sounds like you’re referring to “Men into Space,” a one-season show produced by Ziv. But it starred not Basehart, but William Lundigan (who really does resemble Basehart now that I think about it). The series was done in a fairly serious manner, showing the progress in space travel and eventually setting up a post on the moon. Heavy emphasis on science, not melodrama. Nice little series, actually, although probably too dry for modern tastes.
That was IT! I’ve been trying to recall that show for decades. LOL! THANKS!