We present measurements of the magnetoconductance of quasi-one-dimensional wires fabricated on a two-dimensional electron system in an InSb/InAlSb heterostructure. The width and temperature dependence of the spin and phase coherence lengths in the narrow wires are examined by analyzing the magnetoconductance in antilocalization theory, modified to account for ballistic transport. The experiments indicate that the confined geometry can enhance spin coherence lengths in systems not in the motional narrowing regime and in the presence of strong cubic Dresselhaus spin-orbit interaction. Experimentally, the spin coherence lengths are found to be inversely proportional to wire width and to display a weak temperature dependence. For all wire widths the phase coherence length, after correction for finite length effects, shows a temperature dependence indicative of phase decoherence via the one-dimensional Nyquist mechanism.