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The recently discovered antiferromagnetic (AFM) topological insulator (TI) MnBi2Te4 represents a versatile material platform for exploring exotic topological quantum phenomena in nanoscale devices. It has been proposed that even-septuple-layer (even-SL) MnBi2Te4 can host helical hinge currents with unique nonlocal behavior, but experimental confirmation is still lacking. In this work, we report transport studies of exfoliated MnBi2Te4 flakes with varied thicknesses down to the few-nanometer regime. We observe giant nonlocal transport signals in even-SL devices when the system is in the axion insulator state but vanishingly small nonlocal signal in the odd-SL devices at the same magnetic field range. In conjunction with theoretical calculations, we demonstrate that the nonlocal transport is via the helical edge currents mainly distributed at the hinges between the side and top/bottom surfaces. The helical edge currents in the axion insulator state may find unique applications in topological quantum devices.