You should really watch this video by Bill Hammack. In it, he explains how backlit LCD monitors work (he is literally like Bill Nye the Science Guy, but for adults - his videos are awesome to be honest).
Most modern LCD panels should not "leak" anything, since although liquid crystals are a liquid, they don't necessarily "flow out" of the monitor and leak everywhere. Very rarely will you see actual liquid come out of a broken LCD screen (it is very viscous).
A common chemical to use in LCD screens is MBBA. From this MSDS, you can see that while the material is poisonus if you ingest/inhale it (or get it in an open wound or your eyes), it just causes some skin irritation if you get it on you. If you do happen to get the material inside of your body, then you should seek medical assistance (as it will begin to make methemoglobin in your blood). However, if you get just a bit on your finger (externally only), wash your hands with soap and water and you should be fine.
So, to answer your question, I would not consider it safe to handle, nor would I willingly be around the stuff, so you should avoid it. That being said, if you happen to break an LCD panel and get some on you, throw it out immediately and wash your hands.