Due to the weakness of gravity, the accuracy of the Newtonian gravitational constant G is essentially below the accuracy of other fundamental constants. The current value of G, recommended by CODATA in 2006, based on all results available at the end of 2006, is G = (6.67428 ± 0.00067) × 10-11 m3kg-1s-2 with a relative error of 100 ppm. The modern history of G determination is considered. New experiments at a level of accuracy of 10 to 30 ppm are now in progress in some world gravitational laboratories. One of the problems of improving the accuracy of G is a precision measurement of the period of eigen oscillations of the torsion balance. The torsion balance responses to tiny environmental noises, which accordingly result in changes of the torsional oscillation period. The seismic environmental noise in the underground laboratory condition has been studied on the base of data of the Baksan Laser Interferometer. The dependence of the torsion period value and the error of its estimation on the level of the acting seismic noise has been simulated and studied. To measure the Newtonian gravitational constant at the accuracy level of 10 ppm, the environmental seismic noise must be below 10-2 mGal.