Today, data security is an increasingly hot topic, and thus also the security and reliability of end-user identity verification, i.e. authentication. In recent years, banks began to substitute password authentication by more secure ways of authentication because passwords were not considered to be secure enough. Current legislation even forces banks to implement multifactor authentication of their clients. Banks, therefore, consider using biometric authentication as one of the possible ways. To verify a user's identity, biometric authentication uses unique biometric characteristics of the user. Examples of such methods are facial recognition, iris scanning, fingerprints, and so on. This paper deals with another biometric feature that could be used for authentication in mobile banking applications; as almost all mobile phones have an integrated camera, hand authentication can make a banking information system more secure and its user interface more convenient. Although the idea of hand biometric authentication is not entirely new and there exist many ways of implementing it, our approach based on using convolutional neural networks is not only innovative, but its results are promising as well. This paper presents a modern approach to identifying users by convolutional neural networks when this type of neural network is used both for hand features extraction and bank user identity validation.