As I understand it (see: not at all), if you leave a spaceship with no suit on, you’d get baked like Marie Curie’s ovaries from the radiation. It’s mainly our atmosphere that protects us from most of the nastiest stuff. Would a giant cable reaching from Earth all the way to a platform outside the atmosphere become dangerously-radioactive over time? And if so, would that eventually cause the entire planet to get radioactive over hundreds of years? Kinda like if the hole in the Ozone layer were replaced with a Mario pipe.
And if that is the case, maybe we could forget the elevator aspect of it and just aim for a free eternal source of radioactive energy, like a really shitty Dyson sphere 👀
Induced radioactivity is mostly the result of contamination from radioactive materials. Whilst it’s possible to induce radioactivity from gamma rays directly, you’re talking “background noise” levels of radiation. Which is to say, the cable isn’t going to become notably radioactive, and even then, the part that does, will be the part that isn’t protected by the atmosphere. And for people to navigate those areas of space safely, we already need shielding to protect us from the suns electromagnetic radiation, so a small increase in radiation from the cable isn’t going to make much of a difference to anything.