The interaction between a fluid and a poroelastic structure is a complex problem that couples the Navier-Stokes equations for the fluid with the Biot system for the structure. The finite element approximation of this problem is involved due to the fact that both subproblems are indefinite. In this work, we design residual-based stabilization techniques for the Biot system, that have been motivated using the variational multiscale approach. Then, we state the monolithic Navier-Stokes/Biot system with the appropriate transmission conditions on the interface. We consider both monolithic solvers and heterogeneous domain decomposition strategies. Different domain decomposition methods are used and their convergence has been analyzed for a simplified problem. The efficiency of these methods has been compared for a test problem motivated from hemodynamics applications.