We consider several direct and adjoint Boussinesq static problems under different types of over-determined conditions. We then conclude, in each case, that the solution pair corresponding to {fluid velocity, scalar temperature} must vanish identically on the whole domain, so that the pressure is then constant (Unique Continuation Property). In going from the direct to the adjoint problem, the coupling operators between the fluid and the thermal equations switch places. As a result, the adjoint Boussinesq system has a more favorable structure than the direct Boussinesq system and hence yields UCP results under weaker requirements; typically, a reduction by one or even two units on the number of components of the fluid vector being involved in the assumptions. To illustrate: in the key direct Boussinesq problem, over-determination consists of the additional vanishing of the solution pair in a common arbitrarily small subset of the interior. In contrast, in the corresponding adjoint Boussinesq problem, only the first $$(d-1)$$ components of the d-dimensional fluid velocity vector need to be assumed as vanishing on the interior subset. These UCPs for the adjoint problem are critical ingredients in the solution of corresponding uniform stabilization problems of (direct) dynamic Boussinesq systems by suitable finite dimensional feedback controls. They allow one to verify a corresponding Kalman algebraic condition for controllability.