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, so it wouldn’t be any different on the space elevator.