The peculiar features in the manifestation of the properties of topological insulators (TIs) and quantum size effects on the longitudinal magnetoresistance (LM) (H ||I) of Bi–17 at % Sb single-crystal semiconductor wires prepared by liquid phase casting (Ulitovsky method) having orientation  along the axis and diameters of 75–1000 nm have been studied. At high temperatures (T > 50 K), the quantum size effect is evident as an increase in the energy gap of ΔE ~ d^–1 of the Bi-17 at % Sb semiconductor wires with a decrease in the wire diameter d. A decrease in temperature (T < 50 K) leads to a transition from the semiconductor dependence R(T) to the metallic dependence with a decrease in the wire diameter d; the transition indicates the presence of the surface states characteristic of TIs. In a diameter range of 200–350 nm, the Shubnikov de Haas (SdH) oscillations are observed on the longitudinal magnetoresistance H ||I in weak magnetic fields (H < 3 T); they are used to calculate the Dingle temperature, cyclotron masses, carrier mean free path, and charge carrier mobilities μ = 11 × 10³ cm²/s. On the LM of the Bi–17 at % Sb wires at 4.2 K, a singularity in the form of phase shift of the Landau levels’ index on the SdH oscillations and anomalous maximum in the thickness dependence of the LM at 4.2 K is observed; it is associated with the semiconductor-metal transition due to a significant contribution of the surface states of the TI to the conductivity. Taken together, the singularities on the LM, the phase shift of the SdH oscillations, high mobility and anisotropy of charge carriers, and an increase in conductivity with a decrease in the wire diameter d indicate the presence of surface states in thin semiconductor Bi_{1 – x}Sb_x wires with a Fermi energy of the “Dirac cone” type, which are highly sensitive to the wire diameter, temperature, and the magnitude and direction of the magnetic field and lead to new peculiar features of the transport properties of topological insulators in the low-dimensional structures.