The hydrostatic balance or hydrostatic equilibrium is one of the most prominent topics in fluid dynamics, atmospherics and astrophysics. In fact, our atmosphere is mainly in hydrostatic equilibrium, between the upward-directed pressure gradient force and the downward-directed force of gravity. Understanding the stability of perturbations near the hydrostatic balance of the Boussinesq system may help gain insight into some weather phenomena. This paper is concerned with the 2D anisotropic Boussinesq equations involving only vertical dissipation and horizontal thermal diffusion. When the spatial domain is $\Omega=\mathbb{T}\times\mathbb{R}$ with $\mathbb{T}= [0, 1]$ being a 1D periodic box, this work establishes the stability and specifies the precise large-time behavior of perturbations near the hydrostatic equilibrium. The approach here is to distinguish the vertical averages of the velocity and temperature from their corresponding oscillation parts.