Neutrinos can interact with my quarks (and so the nucleus) and electrons. The energy transferred however is very small. Supernova neutrinos are at around 10-12MeV. A photon with the same energy would be around 0.1mm wavelength, or 2.9Ghz. For comparison, you need a wavelength of around 120nm (2500Thz) to crack DNA. This is in the UV band.

In short, neutrinos would damage via thermal effects. There would be effectively no shielding by other tissue, leading to near perfectly even heating. Your spaceship will also be kicked by the force. However, it would be evenly spread over every atom present. You wouldn't experience any g forces, just the feeling of being cooked alive, perfectly evenly. An external observer would see you get pushed back.

I'm now wondering on the kitchen utility of a neutrino microwave. It would deliver perfectly even cooking, unlike an oven, or microwave oven... The safety shielding and power requirements would likely be non-viable, however.