This paper is concerned with a class of fractional Schr\”{o}dinger equation with Hardy potential \begin{equation}\nonumber (-\Delta)^{s}u+V(x)u-\frac{\kappa}{|x|^{2s}}u=f(x,u),~~x\in \mathbb{R}^{N}, \end{equation} where $s\in(0,1)$ and $\kappa\geq0$ is a parameter. Under some suitable conditions on the potential $V$ and the nonlinearity $f$, we prove the existence of ground state solutions when the parameter $\kappa$ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about $\kappa$. Finally, we are able to explore the asymptotic behaviors of ground state solutions as $\kappa$ tends to $0$.