Distributed acoustic sensing (DAS) has attracted much attention in seismic data acquisition because of its low cost, anti-electromagnetic interference, and high acquisition density. Unfortunately, the acquired DAS records are usually accompanied by various kinds of complex noise, affecting subsequent interpretation and inversion. Traditional methods have difficulties in effectively attenuating the intense background noise. In general, the denoising task of DAS data is challenging. Recently, convolutional neural networks (CNNs) exhibit a good ability in suppressing the noise in DAS records. However, traditional CNN-based frameworks always have a relatively simple network architecture, bringing negative impacts on the denoising capability. To solve this problem, we propose a dual-branch dense network (DBD-Net) in this paper. Specifically, DBD-Net introduces a novel combination of dual-branch modules and an attention mechanism: the dual-branch modules extract multi-scale coarse-to-fine features, while the attention mechanism highlights the most informative features. This joint design strengthens feature representation and signal recovery compared with conventional CNN structures such as denoising CNN (DnCNN) and U-Net. Moreover, an attention module is employed to enhance the effective features. To verify the denoising ability, we compare DBD-Net with other competing methods, including band-pass filter, DnCNN, and U-Net, in terms of denoising capability and processing accuracy. Experimental results verify that DBD-Net can improve the quality of DAS records with a signal-to-noise ratio increment of nearly 26 dB. Meanwhile, the intense DAS background noise is also perfectly suppressed and the weak signals are effectively restored, representing advantages over the competing methods.