By training I am a radiation physicist and actually do know about particle beams hitting people. The LHC beam would certainly give someone an appreciable radiation dose, although the secondary synchrotron X-rays would probably be worse than the proton beam itself. In fact a much lower energy proton beam would have a worse effect on your hand/body, since that beam would actually stop inside you (protons deposit most of their dose at the very end of their range).

Of course the injury you received would largely be determined by the length of time your hand was in the beam. Really not fun results can be radiation ulcers or radiogenic cancer. If the dose were high enough you could get a radiation syndrome, where some/all of the fast growing cells in your body die off and you die in a few days or months.

The only data I could find after a quick search indicated that the dose present in the beam line would be on the order of 10^4 Gy per year. That means if you managed to put your hand in the beam for 1 minute, you're only looking at about 0.01 Gy, which isn't that much.