The constitutive equations of strain gradient plasticity were applied to numerically analyze the combined effect of material plastic properties, the type of stresses, and the Taylor parameter of material structure on the stress fields and dislocation densities at the crack tip. It was found that taking into account the influence of the intrinsic length scale parameter leads to different levels of increase in stresses at the crack tip under plane stress and plane strain in comparison with the solution of classical plasticity. Based on the finite element solutions of strain gradient plasticity theory, the dislocation density fields were found as a function of the distance to the crack tip. The coordinate of the equality of the dislocation density components is assigned to the outer boundary of the dominance region of strain gradient plasticity. The plane stress condition was shown to cause higher values of equivalent stresses and dislocation densities as compared to plane strain under equal loading conditions. As a result of the complex parametric study, the combined effect of plastic properties and Taylor parameter of material structure was assessed in conditions of strain gradient plasticity.