Computer vision syndrome causes vision problems and discomfort mainly due to dry eye. Several studies show that dry eye in computer users is caused by a reduction in the blink rate and an increase in the prevalence of incomplete blinks. In this context, this article introduces Eye-LRCN, a new eye blink detection method that also evaluates the completeness of the blink. The method is based on a long-term recurrent convolutional network (LRCN), which combines a convolutional neural network (CNN) for feature extraction with a bidirectional recurrent neural network that performs sequence learning and classifies the blinks. A Siamese architecture is used during CNN training to overcome the high-class imbalance present in blink detection and the limited amount of data available to train blink detection models. The method was evaluated on three different tasks: blink detection, blink completeness detection, and eye state detection. We report superior performance to the state-of-the-art methods in blink detection and blink completeness detection, and remarkable results in eye state detection.