We report magnetotransport properties of BaZnBi$_{2}$ single crystals. Whereas electronic structure features Dirac states, such states are removed from the Fermi level by spin-orbit coupling (SOC) and consequently electronic transport is dominated by the small hole and electron pockets. Our results are consistent with three dimensional (3D) but also with quasi two dimensional (2D) portions of the Fermi surface. The spin-orbit coupling-induced gap in Dirac states is much larger when compared to isostructural SrMnBi$_{2}$. This suggests that not only long range magnetic order but also mass of the alkaline earth atoms A in ABX$_{2}$ (A = alkaine earth, B = transition metal and X=Bi/Sb) are important for the presence of low-energy states obeying the relativistic Dirac equation at the Fermi surface