We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $\Omega \times \R^3$, for a smooth bounded domain $\Omega$ of $\R^3$, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to $0$ while the distribution function concentrates towards a Dirac mass in velocity centered at $0$, with an exponential rate. The proof, which follows the methods introduced in \cite{HKMM}, requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.