The enhanced ionization(EI) behaviour of linear multiatomic molecular ions is studied in two-color(fundamental radiation:780 nm, the second harmonic:390 nm) laser fields by the numerical solution of the time-dependent Schrdinger equation with the symmetrical splitting of the short-time exponential propagator. The influence of the relative phase between the two-color laser fields on the ionization probability is given. The numerical results demonstrate that the influence of the relative phase on the ionization probability is the strongest in the range of the inter-nuclear distance where the EI occurs. The influence can be explained in terms of the field-induced over-the-barrier ionization model.