The characteristics of fundamental and mutipole dark solitons in the nonlocal nonlinear couplers are studied through numerical simulation in this work. Firstly, the fundamental dark solitons with different parameters are obtained by the Newton iteration. It is found that the amplitude and beam width of the ground state dark soliton increase with the enhancement of the nonlocality degree. As the nonlinear parameters increase or the propagation constant decreases, the amplitude of the fundamental dark soliton increases and the beam width decreases. The power of the fundamental dark soliton increases with the nonlocality degree and nonlinear parameters increasing, and decreases with the propagation constant increasing. The refractive index induced by the light field decreases with the nonlocality degree increasing and the propagation constant decreasing. The amplitudes of the two components of the fundamental dark soliton can be identical by adjusting the coupling coefficient. These numerical results are also verified in the case of multipole dark solitons. Secondly, the transmission stability of fundamental and mutipole dark solitons are studied. The stability of dark soliton is verified by the linear stability analysis and fractional Fourier evolution. It is found that the fundamental dark solitons are stable in their existing regions, while the stable region of the multipolar dark solitons depends on the nonlocality degree and the propagation constant. Finally, these different types of dark dipole solitons and dark tripole solitons are obtained by changing different parameters, and their structures affect the stability of dark soliton. It is found that the multipole dark soliton with potential well is more stable than that with potential barrier. The refractive-index distribution dependent spacing between the adjacent multipole dark solitons favors their stability.