Detailed magnetization, specific heat and Li7 nuclear magnetic resonance (NMR) measurements on single crystals of the hyperhoneycomb Kitaev magnet $\beta$-Li2IrO3 are reported. At high temperatures, anisotropy of the magnetization is reflected by the different Curie-Weiss temperatures for different field directions, in agreement with the combination of a ferromagnetic Kitaev interaction (K) and a negative off-diagonal anisotropy ($\Gamma$) as two leading terms in the spin Hamiltonian. At low temperatures, magnetic fields applied along a or c have only a weak effect on the system and reduce the N{\'{e}}el temperature from 38 K at 0 T to about 35.5 K at 14 T, with no field-induced transitions observed up to 58 T on a powder sample. In contrast, the field applied along b causes a drastic reduction in the TN that vanishes around Hc=2.8T, giving way to a crossover toward a quantum paramagnetic state. Li7 NMR measurements in this field-induced state reveal a gradual line broadening and a continuous evolution of the line shift with temperature, suggesting the development of local magnetic fields. The spin-lattice relaxation rate shows a peak around the crossover temperature 40 K and follows power-law behavior below this temperature.