The homogeneous linewidths of excitons in wet-etched ZnSe quantum wires of lateral sizes down to 23 nm are studied by transient four-wave mixing. The low-density dephasing time is found to increase with decreasing wire width. This is attributed mainly to a reduction of electron-exciton scattering within the wire due to the electron trapping in surface states and exciton localization. The exciton-exciton scattering efficiency, determined by the density dependence of the exciton dephasing, is found to increase with decreasing wire width. This is assigned to the reduced phase space in a quasi-one-dimensional system, enhancing the repulsive interaction between excitons due to Pauli blocking.