The answer is : not much due to the experiment.
The experiment is about making particles reach very high energy, and then smash it together. The highest energy reached in CERN, according to wikipedia is $13 \cdot 10^{12}$ eV (eV stands for electron-volt).
However, cosmic ray (energetic particle coming from the space) hit the earth constantly. Here is a little graph from wikiepedia which explain how much, relative to the energy of the particle :

(the flux of particle is graduated in what could look as a strange unit, but the label on the curve are way more informative, $1$ m$^{-2}$ s$^{-1}$ for example represents the level corresponding to 1 particle crossing an area of 1 meter squared each second)
You can see that around one particle with energy about $10^{19}$ eV pass through a surface (on earth) of one kilometer squared each year (1 km$^{-2}$ yr$^{-1}$ label). That is an energy one billion time hight than is CERN, and given the total surface of earth and its age, if something could have gone wrong with particle of this energy, it would most likely already have happened.
Note however that since in quantum mechanics is about probabilities, worst case scenario in CERN can not be excluded for sure, but it could as well happen inside my kitchen due to a nasty cosmic ray.
So the worst thing which could probably happen at the CERN is some kind of "regular" incident, like fire or leaks of supercooling liquids or gases (if you visit CERN they actually tell you what to do if it happens). Note that the link I gave, reports a leak of "approximately 6 tonnes of liquid helium".