What are the circumstances in which neutrinos can harm humans or even kill them.?
Under regular circumstances, neutrinos are absolutely harmless due to their small interaction cross section. If you hold your thumb up to the sun, about 60 billion (60 · 109) neutrinos will pass through your thumbnail per second, but you will rarely ever have any neutrino interaction in your body.
That is for 'regular' neutrino fluxes which you have from the sun. However, if you take these fluxes and increase them by several orders of magnitude, you can get into a dangerous area. Martin Beckett already mentioned supernovas, but one potentially realistic hazard comes from muon storage rings. I've found a paper that discusses this:
Muons decay into electrons and neutrinos ($\mu^- \rightarrow e^- + \bar \nu_e + \nu_\mu$), which would fly in the direction of the neutrino beam, so tangentially to the beampipe. The neutrinos eventually interact with matter, and produce harmful secondary radiation. If some day someone would build a 10 TeV muon collider, the author estimates an annual dose of 0.66 mSv in the plane of the ring, which is above the US legal off-site radiation limits (according to the linked paper). In long, straight sections of the accelerator this gets worse, since the neutrinos accumulate in a fairly small angular section.
However, this does not mean that a muon collider will produce an all-penetrating death ray in it's plane of operation. Remember, the thing that makes radiation dangerous is its energy loss in matter, and that's the same effect that makes it shieldable. You can build it further underground, tune its shape, etc., to make it less dangerous.
One small anecdote at the end: The MINOS experiment at Fermilab, one of which's goals was to study neutino oscillations, used to send a neutrino beam from the Tevatron's Main Injector through its "near detector" (at Fermilab, near Chicago) to the "far detector" (in Minnesota). They were looking for neutrino (dis-)appearance due to oscillations. When they had visitors, they used to take them on a tour of the near detector hall, even when the beam was running. Supposedly, the detector was sensitive enough (and their beam intensive enough) that if someone would stand in front of it for too long, they would leave a shadow in the detector :-). I'm not sure how long exactly they would have to stand there, though it was apparently harmless.
Since neutrinos interact so rarely with matter (ie you) you would need an awful lot of them to have any significant effect. A supernova conveniently emits about 99% of its energy as neutrinos. So if you were standing quite close to a supernova when it went off then there could possibly be enough interactions to harm you.
If you stay a respectable distance from any exploding stars then you are OK
But SN1987a was around 160,000lyr away. If you were at the distance of Earth's orbit from the star (15$\mu$ light years away) you would have received (160,000/0.000015)^2 = 10^20 as many neutrinos going through you.
I don't know what the biological effects of each neutrino interaction is (I doubt it's been the subject of many clinical trials) but having 2.5 x10^16 of them stop in your body is unlikely to be healthy.
Although it's worth pointing out that if you are standing 1 AU from a supernova this is probably the least of your worries.