Both the (current) answers to the question state that it is not possible for your idea to work. Aside from the fact that you are explicitly not asking for a feasibility study, the obligatory XKCD reference shows that it's all about the solid angle of the detector from the supernova and so, such a detector may indeed be feasible far into the future. In addition, this answer ignores a number of annoying details in the hope of getting a decent approximation to an answer relatively quickly and easily. It's not a [tag:hard-science] question, after all!
It turns out that, after making these various estimates and approximations, the efficiency/average energy collected per antineutrino incident on the detector from a supernova would be around $5.3\times 10^{-26}\,\text{J}$.
Background
To start with, let's note the fundamental principles behind what's going on behind <PERSON> style neutrino detection:
1. <PERSON> radiation: Similar to a sonic boom, when a charged particle travels through a medium with refractive index $n$ and travels faster than the speed of light in that medium, $\frac{c}{n}$, (in the case of standard materials with $n>1$), it emits light, typically detected by lots of photomultiplier tubes. This is commonly observed in water, which has a refractive index of $n \approx \frac{4}{3}$ (for the purposes of this answer, this is a good enough estimate for wavelengths of light $\lesssim 17.5\,\mu m$).
2. Detecting neutrinos: Neutrinos are uncharged, so cannot cause <PERSON> radiation by themselves. As such, some sort of interaction needs to occur in order to get a charged particle that can in turn cause radiation. This is generally done by the weak interaction. 70
The most common example in a <PERSON> detector is an $\bar\nu_e+ p \rightarrow n + e^+$. That is, an anti-neutrino converts a proton (Hydrogen) into a neutron and emits a positron in the process. The positron then emits <PERSON> radiation before getting absorbed by a similar process at some later point.
3. Detecting/using the emitted photons: Once generated, a photon has to be able to go from the point of generation to the point of detection. Of course, photons can also interact with and get lost in the surrounding environment. This is described by the transmittance, $T$, of light through the material, which is (in terms of the attenuation coefficient, $\alpha$, over a distance $d$) $T = e^{-\alpha d}$. For light with an initial intensity $I_0$, the final intensity is then $I_0T$, which can then be detected.
However, there are some Issues with the Standard Model of particle physics: (all of these will be ignored as the definite model for how these are actually fixed is obviously unknown, but are here for completion)
* It doesn't account for gravity (any effects of this on neutrinos are, to my knowledge, unknown). As this has effects on large energy collisions, which could occur in supernovae, this may have en effect on the production of neutrinos and antineutrinos.
* The Standard Model predicts that neutrinos are massless, yet the 2015 Nobel Prize was awarded for experimentally showing that neutrinos do in fact have mass This would have an effect on the scattering cross-section of neutrinos but as we don't yet know what the correct model is1, this can't really be accounted for here.
* Recent particle physics experiments at e.g. CERN have begun to indicate physics beyond the standard model (note: this is not conclusive evidence by some margin, but is a good indication). It isn't impossible that this would have an effect on neutrinos.
Calculating the Efficiency
Cross section of the Interaction
the cross section for the relevant type of neutrino scattering is2 $$\sigma = \frac{4G^2_FM_W^2E_{e^{+}}^2}{\pi\left(M_W^2 + 4E_{e^{+}}^2\right)},$$ where (in natural units) the Fermi constant, $G_F \approx 1.16\times 10^{-5}\,\text{GeV}^{-2}$ and the mass of the weak boson, $M_W \approx 80.4\, \text{GeV}$. 70