One of the significant problems of streaming data classification is the occurrence of concept drift, consisting of the change of probabilistic characteristics of the classification task. This phenomenon destabilizes the performance of the classification model and seriously degrades its quality. An appropriate strategy counteracting this phenomenon is required to adapt the classifier to the changing probabilistic characteristics. One of the significant problems in implementing such a solution is the access to data labels. It is usually costly, so to minimize the expenses related to this process, learning strategies based on semi-supervised learning are proposed, e.g., employing active learning methods indicating which of the incoming objects are valuable to be labeled for improving the classifier's performance. This paper proposes a novel chunk-based method for non-stationary data streams based on classifier ensemble learning and an active learning strategy considering a limited budget that can be successfully applied to any data stream classification algorithm. The proposed method has been evaluated through computer experiments using both real and generated data streams. The results confirm the high quality of the proposed algorithm over state-of-the-art methods.