Bogolyubov quasiparticles move in a practically uniform magnetic field in the vortex state of high-temperature cuprate superconductors. When set in motion by an externally applied heat current, the quasiparticles' trajectories may bend, causing a temperature gradient perpendicular to the heat current and the applied magnetic field, resulting in the thermal Hall effect. Here we relate this effect to the Berry curvature of quasiparticle magnetic sub-bands, and calculate the dependence of the intrinsic thermal Hall conductivity on superconductor's temperature, magnetic field and the amplitude of the d-wave pairing. The intrinsic contribution to thermal Hall conductivity displays a rapid onset with increasing temperature, which compares favourably with existing experiments at high magnetic field on the highest purity samples. Because such temperature onset is related to the pairing amplitude, our finding may help to settle a much-debated question of the bulk value of the pairing strength in cuprate superconductors in magnetic field.