How did "Secure Voice" work on the old Air Force One's? I recently visited the "Museum of Flight" where they have an old SAM 970, previously known as "Air Force One".

Inside there is a comm station with many buttons designated "Secure Voice", as can be seen on the lower row of controls in the attached picture. The plaque says "During the Johnson and Nixon eras, "Air Force One" was equipped with a radiophone on which secret voice messages could be scrambled and sent. It rarely functioned properly and was removed to save weight." Therefore I suspect that the buttons and controls in question is not related to this radiophone.

Does anybody know what kind of Secure Voice communications they used these controls for, and how these works?

As a bonus, if the radiophone that rarely worked could be explained as well, I'd really appreciate it :)
They used a scrambler[wikipedia] (not “encryption“ and definitely not a “One-Time-Pad” as some think [comment] [comment]) to transmit messages via a SatCom (read: military satellite) system. The scrambler they used back then was (let's just call it) “non-optimal” and has been replaced with an enhanced system. That's also what the sign is hinting at, among other things.

Scramblers and jam resisting implementations are only distantly related to cryptography, but somewhat comparable to how synchronizable, bit-wise stream ciphers and/or lightweight hash algorithms work. Yet, the goal of scramblers is different from a security perspective.

Scramblers use a secret spreading sequence or secret hop sequence, or some other information that must be kept secret from a potential jammer and/or listener. This mostly (and certainly back then) assumed sender and receiver to have a shared secret… somewhat like a seed for an (CS)PRNG. Back in the days, those shared secrets were stored analogue, while meanwhile things have (in most cases) been digitalized.

As a matter a fact: at the time of writing this, there is only one scrambling system for achieving jam resistance without any shared secret: BBC coding. BBC coding is able to provide unkeyed jam resistance. For a quick read on the subject, you can check (for example) the MilCom2012 paper The Glowworm hash: Increased Speed and Security for BBC Unkeyed Jam Resistance (PDF).

I'll skip the “Bonus” about “the radiophone that rarely worked“ as that would push this Q&amp;A too much into the off-topic area and will quickly become too broad. Instead, I'll add an aside…

Aside: In case you'ld like to dive into the current state of scrambling and jam resistance, you could take a look at a related meta question I once asked, which happens to contain a whole list of related papers. Using your favorite search engine, you'll be able to find the related papers as PDFs.