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Perivascular spaces (PVS) in the human brain are related to various brain diseases. However, it is difficult to quantify them due to their thin and blurry appearance. In this paper, we introduce a deep-learning-based method, which can enhance a magnetic resonance (MR) image to better visualize the PVS. To accurately predict the enhanced image, we propose a very deep 3D convolutional neural network that contains densely connected networks with skip connections. The proposed networks can utilize rich contextual information derived from low-level to high-level features and effectively alleviate the gradient vanishing problem caused by the deep layers. The proposed method is evaluated on 17 7T MR images by a twofold cross-validation. The experiments show that our proposed network is much more effective to enhance the PVS than the previous PVS enhancement methods.