Where do you think uranium and other radioactive minerals come from? Underground. From a long time ago.
Uranium ore is not nearly as radioactive as the enriched fuel used in the powerplants.
It's win-win. We get rid of the waste, and in the off-chance there are evil aliens on Mars we are doing the modern-day equivalent of catapulting a dead, disease-ridden cow over the city wall to slowly kill those inside.
But what if they aren't evil? You wouldn't want to take the risk of slowly killing brain meltingly cute puppy-eyed space kittens, would you :teehee:
Seriously, space elevators are still science fiction. And even then there are things that could go wrong, a launch gone wrong and you have a high atmosphere radiation bomb.