Due to high spatial resolution, low cost, and wide bandwidth, distributed optical fiber acoustic sensing (DAS) is regarded as a potential tool for data acquisition in vertical seismic profile (VSP) surveys. However, in real DAS-VSP records, desired signals are often seriously plagued by various noise, which does not appear in the conventional seismic data received by electronic geophones. Exploring a high-performing attenuation method for the background noise can significantly improve the quality of DAS-VSP records and has essential impacts on the following imaging and interpretation. Deep-learning-based methods, especially convolutional neural network (CNN), have shown remarkable performance in seismic data denoising. However, the conventional CNN-based methods may degrade when dealing with DAS-VSP records in low signal-to-noise ratio. In this study, we propose a novel multi-scale dense-connection denoising network (MDD-Net) to achieve high-accuracy processing of the complex DAS background noise. Unlike conventional multi-scale networks, MDD-Net utilizes widen convolution block to capture the multi-scale features of the analyzed data. On this basis, dense connection operations are employed to fuse the features and improve the network efficiency. Meanwhile, an enhanced spatial attention (ESA) block is designed to reinforce the features, which are helpful for noise suppression and weak signal recovery. Both synthetic and field DAS-VSP records are processed to verify the effectiveness of MDD-Net. Meanwhile, we also compare the denoising results with other competing methods. The experimental results demonstrate that MDD-Net can significantly attenuate the complex DAS background noise and restore the desired signals, even for the weak upgoing signals.