I've seen two shows at the Old Globe and had a great time during each visit. Both shows have been in the enclosed theatre, which has a distinct 60s/70s vibe on the interior, but it doesn't take away from the productions. Seats are comfortable and the house is set up so that there really aren't any bad seats (and their online ticketing process is great-- you can actually choose to see what the stage will look like from the seat you pick). The theatre grounds are beautiful and the buildings evoke the British theatre from which it takes its name.
Their productions are top-notch and if you get to the theatre early, you can take in some of Balboa Park first, making for a perfect outing.