I guess there is some way that I can detect that something is not quite right, before everyone just dies on me. But how?

If I recall correctly, if you have a Laboratory, you can simply drop a corpse on the... research machine thingamabob and your scientists will perform an autopsy, revealing the cause of death.

Nice! I haven't reached that Laboratory yet, but beat the level anyhow. I always have a bit of trouble noticing disasters in sim games. I think everything's fine until everyone starts dying :-P

It's all about keeping your station clean (so there isn't any litter to attract the hungry, disease-ridden pests called Space Vermin & Memaus*) and only hiring competent Greys (skill level 3 and up) with no criminal record for your sick bays.

*I've noticed that some Peeps have a criminal record for Memaus smuggling (though I've yet to catch them in the act) so I can't guarantee cleanliness prevents them from showing up at your station but at least they won't have anything to eat.

It seems to me that a post-mortem often fails to reveal the actual cause of death though that might depend on the skill level of the Turrakken scientists analysing the body.