I remember that.
I wrote a game show question about it, I drive by the place where it was recorded two or three times a week (the old A&M Studios on La Brea near Sunset, originally built by Charlie Chaplin and now owned by the Jim Henson people) and I particularly remember that after the song was recorded and people were leaving, somebody offered Bruce Springsteen a security escort to his car (he had parked on the street) and he said, "No, thanks, I'll be all right."
Weird thing is, I don't feel a day over 40.
--t