Perivascular spaces (PVS) in the human brain are related to various brain diseases or functions, but it is difficult to quantify them in a magnetic resonance (MR) image due to their thin and blurry appearance. In this paper, we introduce a deep learning based method which can enhance a MR image to better visualize the PVS. To accurately predict the enhanced image, we propose a very deep 3D convolutional neural network which contains densely connected networks with skip connections. The densely connected 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 seventeen 7T MR images by a twofold cross validation. The experiments show that our proposed network is more effective to enhance the PVS than the previous deep learning based methods using less layers.