Radiation is one of those “gonna’ have to worry about it” issues for Space Elevator passengers (and possibily some of the cargo), but one that can, hopefully be dealt with. If I’m in a climber, I would like some sort of positive reinforcement that whatever shielding has been provided is working and that I’m not getting pinged with too much radiation.
The National Space Biomedical Research Institute, in cooperation with faculty and midshipmen from the United States Naval Academy, is developing a portable measuring device for measuring radiation. They’re targeting it for Lunar and Martian missions, but I see no reason why it wouldn’t work on a Space Elevator.