We consider open non-twist Hamiltonian systems represented by an area-preserving two-dimensional map describing incompressible planar flows in the reference frame of a propagating wave, and possessing exits through which map orbits can escape. The corresponding escape basins have a fractal nature that can be revealed by the so-called basin entropy, a novel concept developed to quantify final-state uncertainty in dynamical systems. Since the map considered violates locally the twist condition, there is a shearless barrier that prevents global chaotic transport. In this paper we show that it is possible to determine the shearless barrier breakup by considering the variation of the escape basin entropy with a tunable parameter.