We present a new model reduction technique for steady fluid-structure interaction problems. When the fluid domain deformation is suitably parametrized, the coupling conditions between the fluid and the structure can be formulated in the low-dimensional space of geometric parameters. Moreover, we apply the reduced basis method to reduce the cost of repeated fluid solutions necessary to achieve convergence of fluid-structure iterations. In this way a reduced order model with reliable a posteriori error bounds is obtained. The proposed method is validated with an example of steady Stokes flow in an axisymmetric channel, where the structure is described by a simple one-dimensional generalized string model. We demonstrate rapid convergence of the reduced solution of the parametrically coupled problem as the number of geometric parameters is increased.