Neutrinos are not directly detected. What happens is that they interact with matter in a medium and produce muons or electrons that are moving faster than the speed of light in that medium. That's only possible because the speed of light in said medium is less than the speed of light in vaccuum by the way. These fast-moving, charged particles lead to the generation of a cone of cherenkov radiation, not unlike a supersonic object is producing a mach cone. From that cone, the direction of the particle motion can be inferred.
One could also keep the detector running while switching the source on and off to establish a baseline and calculate the excess of detected particles while the source was switched on.
Indeed it’s the child-particles of the interactions that are observed, so, electrons muons and Tau particles. Depending how you are measuring, (many approaches) these interactions are generally separable - and usually you can get some directionality out of your measured event - so you know the direction it came from.
Combining the direction with timing - with an accelerator you know exactly when the pulse of neutrinos were produced - this means you can exclude a very large amount of background.
Some background does get through, but you can measure it with e.g. the source turned off as you suggest, or the times when you know the source isn’t otherwise producing (like in between pulses) to get a background rate. You then generate enough data from the source to effectively “drown out” the background.
Much the same can be done with reactor sources - very often when building an experiment with a reactor as a source, there will be some government-level agreement to allow knowledge of what power the reactor is being run at, as well as shutdowns.
Along with Cherenkov, there are and have been scintillator, chemical-energy-deposition, Photographic film and even ocean-audio based methods of detection. Probably some more exotic ones I am forgetting also.
(I did a PHD in neutrino physics, although been out of the field a few years)
I did not know that "ocean-audio based methods of detection" for neutrinos are even possible and will certainly look that up later.
It always seemed a neat idea, that you could instrument absolutely gigantic volumes with.