  In this paper, we investigate the asymptotic stability of the
three-dimensional Couette flow in a stratified fluid governed by the
Stokes-transport equation. We observe that a similar lift-up effect to the
three-dimensional Navier-Stokes equation near Couette flow destabilizes the
system. We find that the inviscid damping type decay due to the Couette flow
together with the damping structure caused by the decreasing background density
stabilizes the system. More precisely, we prove that if the initial density is
close to a linearly decreasing function in the Gevrey-$\frac{1}{s}$ class with
$\frac{1}{2}< s\leq 1$, namely,
$\|\varrho_{\mathrm{in}}(X,Y,Z)-(-Y)\|_{\mathcal{G}^{s}}\leq \epsilon$, then
the perturbed density remains close to $-Y$. Moreover, the associated velocity
field converges to Couette flow $(Y, 0, 0)^{\top}$ with a convergence rate of
$\frac{1}{\langle t\rangle^3}$.
