A method is introduced that establishes relations between properties of collective excitations in finite nuclei and the phase transition density nt and pressure Pt at the inner edge separating the liquid core and the solid crust of a neutron star. A theoretical framework that includes the thermodynamic method, relativistic nuclear energy density functionals, and the quasiparticle random-phase approximation is employed in a self-consistent calculation of (nt,Pt) and collective excitations in nuclei. Covariance analysis shows that properties of charge-exchange dipole transitions, isovector giant dipole and quadrupole resonances, and pygmy dipole transitions are correlated with the core-crust transition density and pressure. A set of relativistic nuclear energy density functionals, characterized by systematic variation of the density dependence of the symmetry energy of nuclear matter, is used to constrain possible values for (nt,Pt). By comparing the calculated excitation energies of giant resonances, energy-weighted pygmy dipole strength, and dipole polarizability with available data, we obtain the weighted average values: nt = 0.0955 ± 0.0007 fm−3 and Pt = 0.59 ± 0.05 MeV fm−3. This approach crucially depends on experimental results for collective excitations in nuclei and, therefore, accurate measurements are necessary to further constrain the structure of the crust of neutron stars.