One of my friends told me that some crew members you pick up randomly on your journey can be a mentally unstable and may go on a murderous rampage and attack other crew members. Is there any way to tell if someone is apt to 'snap' at some point?
Many of the effects of encounters are randomized -- even if last time you were in the exact same situation with the same text, things might go differently. You won't be able to tell if the crewmember will be insane until the event completes, except in certain special circumstances (detailed below).
I believe in one particular event, you find a station with a sole survivor, and (s)he offers to join your crew. There's a chance that everything goes fine, and this will add +1 crewmember. However, there's a chance that this person will go insane and murder one of your other crewmembers, in which case you end with -1 crewmembers. The outcome is randomized, though.
The insanity is immediate during the event, and unlike a "boarding party" you can't fight the crewmember. If they go insane and there's no mitigating factors (see below) you will instantly lose a crewmember. If they don't, you will never have problems with the crewmember in the future.
A couple of things can affect the outcome, however:
I believe that certain event has been changed or you did not have a Slug crew member on board at the time. If you have a Slug and you meet a sole survivor the blue option text that is usually positive will show an option that is along the lines of "Get your Slug to examine this man's mind." This will result in either "He is fine sir" in which case you welcome him aboard or "This man is a dangerous, warped individual, we best leave him" and you abandon him. With a Slug aboard this particular event you will you a crewmember or gain no reward, but at least you don't lose your prized Zoltan.