A few distinctions which're important to keep in mind:
A Command Computer can only be directly accessed by OPs in creative (meaning that if your staff
could use one, they wouldn't
need one for their usual tasks). You can give indirect - and rather more importantly
controlled - access by hooking up an Advanced Monitor and rigging the startup script to take input from that. Command Computers are also immune to disk drive shenanigans.
Command Blocks can be controlled via a regular computer, meaning you don't have to create any "external monitor" setups for your staff to use them. Catch is anyone with a redstone torch and physical access to the blocks can
also use them, and if you enable the CC config option which lets you wrap them as peripherals, then anyone who can get near a command block will be able to reprogram it and do pretty much whatever they like!
Command Computers can get text output from commands which provide it (allowing tricks like the one Yami posted above). ComputerCraft
can't get text output from regular Command Blocks - only redstone output.
If you wanted to get the name of the closest player to a Command Computer though, the "xp" command would probably be a better bet than "tp". Personally, I'd build a solid bedrock chamber with a redstone button outside on the wall next to the computer on the inside; when the button is pressed,
check player positions, and if a staff member is closest to the computer then teleport them into the sealed chamber where the access monitor is.
And if they're
not a staff member, perhaps have some fun with them. Teleport them up a hundred blocks or so, perhaps. Above a pool of water. Maybe.
If you're not familiar with coding (eg if little to none of what I've said above makes sense to you, or more to the point, you've no idea how to research any context you're missing without help), then this project sounds like a recipe for disaster - either you can trust your staff, in which case you should straight-up give them OP access, or you can't, in which case one slip-up in your logic and you'll give full power to someone who shouldn't have it.