Kryptonite as seen in movies and comics only affects Kryptonian people at a biological level. However, in the TV series Smallville, people who stole the powers from him or got the powers from a different source still are weakened by Kryptonite even if they aren't Kryptonian. Why is it exactly that it weakens them when they have Superman's power?
It doesn't make sense that it affects humans with Superman's power, they aren't biologically Kryptonian so why would it affect them? Having the powers transfered makes the body just like a Kryptonian is my only guess.