You measure the time it takes to go from one place to another. Just like you would with a car, only you use particle detectors. (For charged particles of known mass and speed less than about 99% of the speed of light you can also measure the relationship between their energy and momentum, but that doesn't apply to neutrinos.)
The speed of light is about $30\,\mathrm{cm/ns}$ (that's one foot per nanosecond in to US physicists) and it is relatively easy to achieve timing accuracy of a few nanoseconds in small detectors so this is a bit of a work-a-day thing for charged particles and is done over distance of a few meters to a few tens of meters.
In the case of neutrinos the low cross-section means that the detectors are relatively large (meters across) and so if you want much precision you have to use a fairly long baseline (meaning a few hundred kilometers).
Not withstanding the hoopla a few years ago at OPERA (which was the result of a faulty cable connection, of all things) no neutrino speed measurement has ever differend significantly from the speed of light.
But that is to be expected as the neutrinos we measure have energies of a few MeV and up, while their masses is none to be rather less than 1 eV. That means the the Lorentz factor is $\gamma > 10^6$.