The fidelity of a speech signal deteriorates severely in a distributed acoustic sensing (DAS) system due to the influence of the random noise. In order to improve the measurement accuracy, we have theoretically and experimentally compared and analyzed the performance of the speech signal with and without a recognition and reconstruction method-based deep learning technique. A complex convolution recurrent network (CCRN) algorithm based on complex spectral mapping is constructed to enhance the information identification of speech signals. Experimental results show that the random noise can be suppressed and the recognition capability of speech information can be strengthened by the proposed method. The random noise intensity of a speech signal collected by the DAS system is attenuated by approximately 20 dB and the average scale-invariant signal-to-distortion ratio (SI-SDR) is improved by 51.97 dB. Compared with other speech signal enhancement methods, the higher SI-SDR can be demonstrated by using the proposed method. It has been effective to accomplish high-fidelity and high-quality speech signal enhancement in the DAS system, which is a significant step toward a high-performance DAS system for practical applications.