In this paper, we first establish the global existence and stability of solutions to 3-D classical Navier-Stokes equations (NS) in an infinite cylindrical domain with large Fourier mode initial data. Then we extend similar result for 3-D anisotropic Navier-Stokes equations (ANS). We remark that due to the loss of vertical viscosity in (ANS), the construction of the energy functionals for (ANS) is much more subtle than that of (NS). Compared with our previous paper for (NS), we improve the polynomial decay in k for the Fourier coefficients of the solution to be exponential decay in k here.