In short, the Voyager spacecraft were designed (after considerable design and operational experience from Mariner and Pioneer spacecraft) to be able to operate for a long time largely automatically without too much hand-holding. All major systems are multiply redundant and may be remotely turned on and off. While there has been personnel turnover on the program, it has not been of a magnitude to jeopardize program continuation. Finally, program management has been media savvy and well politically connected ensuring that operations are still funded. (Contrast with other missions such as Magellan to Venus which was deorbited while it still had propellant reserve leaving some portions of the planet unmapped)

JPL has (or had?) a "retiree badge" program that permitted retired staff to continue to access their office. Many programs benefited from highly knowledgeable personnel essentially continuing to report to the office every day without pay (not being paid comes with the luxury of not having to worry about being laid off if your charge accounts don’t have enough funds!) It was an absolute privilege to learn from these people.