Are the insides of LCD monitors dangerous?

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.

I have no specific knowledge regarding the exact chemicals involved here, nor have I ever seen an MSDS for this stuff. Now, that said:

I used to work for a laptop manufacturer doing support, and we were instructed that in the event that a customer ever called reporting a leaking LCD we were to IMMEDIATELY tell them to hang up the phone, call emergency services, and get a hazmat team onsite, while we escalated the call to our internal dedicated safety response team. That seems like a pretty harsh response (it's more severe than what I would have been expected to do for anything short of "My laptop set my house on fire") so I'm inclined to think this stuff isn't especially safe. It may be true that it's only a hazard if ingested, though; based on my semi-limited knowledge of the chemistry involved that seems possible. If it were me, though, I'd take the extra cautious route just in case.

(In practice, this basically never happens short of a severe puncture like a blade would cause because of the way these things are manufactured. I've actually never personally heard of a situation where an LCD was leaking short of taking a bullet.) EDIT: Let me just make this stronger. This hardly ever happens, ever, anywhere, ever - it's designed to prevent exactly this sort of problem. Unless you actually witness it in action there is probably no reason to worry about it. See further discussion in comments.