From the viewpoint of physical-layer authentication, spoofing attacks can be foiled by checking channel state information (CSI). Existing CSI-based authentication algorithms mostly require a deep knowledge of the channel to deliver decent performance. In this paper, we investigate CSI-based authenticators that can spare the effort to predetermine channel properties by utilizing deep neural networks (DNNs). We first propose a convolutional neural network (CNN)-enabled authenticator that is able to extract the local features in CSI. Next, we employ the recurrent neural network (RNN) to capture the dependencies between different frequencies in CSI. In addition, we propose to use the convolutional recurrent neural network (CRNN)---a combination of the CNN and the RNN---to learn local and contextual information in CSI for user authentication. To effectively train these DNNs, one needs a large amount of labeled channel records. However, it is often expensive to label large channel observations in the presence of a spoofer. In view of this, we further study a case in which only a small part of the the channel observations are labeled. To handle it, we extend these DNNs-enabled approaches into semi-supervised ones. This extension is based on a semi-supervised learning technique that employs both the labeled and unlabeled data to train a DNN. To be specific, our semi-supervised method begins by generating pseudo labels for the unlabeled channel samples through implementing the K-means algorithm in a semi-supervised manner. Subsequently, both the labeled and pseudo labeled data are exploited to pre-train a DNN, which is then fine-tuned based on the labeled channel records.