Spatiotemporal modulation instabilities (MI) in nonlinear asymmetric oppositely directional coupler with negative-index metamaterial are investigated. The influences of input power, power ratio of the forward- and backward-propagating waves, and coupling coefficients on the spatiotemporal MI are discussed. It is found that the MI can occur only if the transverse wave numbers satisfy the threshold condition. The results also show that increasing the input power or decreasing power ratio of the forward- and backward-propagating waves promotes the spatiotemporal MI and merges different instability regions together. Moreover, decreasing the coupling coefficient can suppress the peak gain and spread the gain spectrum. The controllable electromagnetic parameters of negative-index metamaterials will provide more methods to engineer the MI and soliton formation in this coupler.