In this work we design algebraic stabilization techniques for compressible gas dynamics. Specifically, the monolithic convex limiting (MCL) framework designed for hyperbolic problems is extended to the viscous case of compressible Navier--Stokes equations (NSE). Compared to hyperbolic conservation laws, for which the MCL technique was first designed, the NSE contain second-order diffusive terms which can be handled by a splitting approach (e.\ g., Strang) or in a monolithic manner. Pursuing the latter, we generalize the previously mentioned limiting techniques to the parabolic regime. To this end, we make use of semi-implicit time stepping schemes. Our target scheme is based on our previously employed dissipation-based WENO-stabilization of standard Galerkin methods. Invoking a shock-detector, these schemes use adaptive dissipation for shock-capturing purposes yielding high-order accurate solutions in smooth regions and resolving discontinuities accurately, too. However, there is no guarantee, that numerical approximations are physically meaningful. To enforce this property, we thus perform failsafe MCL-type limiting on top of WENO-based stabilization to guarantee the validity of nonnegotiable constraints. In the case of the Navier--Stokes equations, these properties are nonnegativity of density and internal energy. In our numerical examples, we apply the newly developed discretizations to common test problems to demonstrate their capabilities. We then draw conclusions and give an outlook to future work.