The effect of Rashba spin-orbit coupling on the transport properties of InGaAs/InP quantum wire structures is investigated. The geometry of the wire structures was defined by selective wet chemical etching. For wires without a gate a clear beating pattern, due to the presence of the Rashba spin-orbit coupling, is observed for wires with a width down to 600 nm. For narrower wires no beating pattern is found. The experimental observations are explained by contribution of the Rashba spin-orbit coupling to the one-dimensional magnetosubbands. By depleting the one-dimensional conductor by means of a gate electrode the Rashba coupling strength could be controlled.