The human voice conveys unique characteristics of an individual, making voice biometrics a key technology for verifying identities in various industries. Despite the impressive progress of speaker recognition systems in terms of accuracy, a number of ethical and legal concerns has been raised, specifically relating to the fairness of such systems. In this paper, we aim to explore the disparity in performance achieved by state-of-the-art deep speaker recognition systems, when different groups of individuals characterized by a common sensitive attribute (e.g., gender) are considered. In order to mitigate the unfairness we uncovered by means of an exploratory study, we investigate whether balancing the representation of the different groups of individuals in the training set can lead to a more equal treatment of these demographic groups. Experiments on two state-of-the-art neural architectures and a large-scale public dataset show that models trained with demographically-balanced training sets exhibit a fairer behavior on different groups, while still being accurate. Our study is expected to provide a solid basis for instilling beyond-accuracy objectives (e.g., fairness) in speaker recognition.