We consider the screening of an impurity moment in a d-wave superconductor under the influence of a Zeeman magnetic field. Using the numerical renormalization group technique, we investigate the resulting pseudogap Kondo problem, in particular the field-induced crossover behavior in the vicinity of the zero-field boundary quantum phase transition. The impurity spectral function and the resulting changes in the local host density of states are calculated, giving specific predictions for high-field scanning tunneling microscopy measurements on impurity-doped cuprates.