I was thinking whether neutrinos (or antineutrinos) could be used to propel space craft. This is because neutrinos are plentiful in the universe. What if neutrinos could interact through the weak force with electrons and this energy could be used to power the spacecraft? Would there be a tiny force, because even though the weak force is weak if there are many neutrinos this could increase the chances of an interaction?
I'm looking for a reality check on this.
Neutrinos have, believe it or not, been proposed as a method of spacecraft propulsion - see Morgan (1999). The idea is a bit different from yours (which I believe has not been explored yet), inasmuch as it relies on the production of neutrinos, rather than using existing ambient neutrinos.
This idea utilizes the decay of muons, heavier cousins of electrons.
First, protons and antiprotons are annihilated together. Some of the particles produced include charged and neutral pions.
Pions themselves are unstable, and so they quickly decay. One major decay product is muons.
Muons, too, are unstable. Initially, they are decelerated and then polarized.
The polarized muons decay. Their decay process involves the emission of a muon neutrino and an electron antineutrino. The weak force does not conserve parity, which in this case means that the emission of neutrinos is anisotropic, resulting in a net thrust in one direction.
The efficiency of the process depends on the number of pions produced by the proton-antiproton annihilation, as well as what fraction of the momentum is carried away by neutrinos. Morgan estimates an efficiency of $\eta\simeq0.025$ from this implementation of the process.
The issue with harnessing the neutrino-electron interaction is the cross-sections involved. Since neutrinos are charge-less, and interact only via the weak force (gravity also but that won't help us here), they are more likely to interact with larger particles. For this reason, neutrinos-nuclei interactions are the most common.
While neutrinos can (and do interact) with electrons, they will likely encounter a much higher number of nucleons first. Here you can find some standard information on the neutrino scattering cross-sections. Note that the lowest nucleon-neutrino cross-sections will be three or more orders of magnitude larger than the electron-neutrino cross-sections.
Electrons don't like to be grouped together (both the electrostatic force and degeneracy pressure will push them apart) so to get a high enough concentration of electrons so that you'll be more likely to have e-$\nu$ interactions will be difficult. To compound this, high enough concentrations of neutrinos, to warrant enough interactions per second to result in energies required for propulsion, cannot be found anywhere except near supernovas and if you're near those, propulsion will be the least of your worries.
For comparison, on the Earth, the highest concentration of neutrinos come from the sun, which is estimated to produce neutrino flux on the order of ~ $10^{11}/cm^{2} s$ (source), whereas the interaction probability for a 1 MeV neutrino is on the order of ~ $10^{-11}$, for interacting with anything in the Earth as it passes through (source).
If you manage to interact with neutrinos to any practical extent, a lot of possibilities open at once:
Energy from the star neutrinos - the Sun emmits some 15% of its energy in neutrinos, you can harvest them even in some planet's shadow. Hotter-core stars are even better in this regard.
Neutrino sails - to harvest neutrino momentum. If you can interact separately to high-energy star neutrinos and with low-energy neutrino background, you can use them both as wind-and-water marine navigation in whatever direction you like.
Neutrino cooling - you can offload some heat from whatever reactor you have to the neutrino background, instead or in addition of radiating it in electromagnetic waves.
