Structured sequences are a popular data representation, used to model complex data such as traffic networks. A key machine learning task for structured sequences is node classification, that is predicting the class labels of unlabeled nodes. Though many node classification models were proposed, they assume a closed world setting, that all class labels appear in the training data. But in the real-world, the presence of never-before-seen class labels in testing data can considerably degrade a classifier's accuracy. A promising solution to this issue is to build classifiers for an open-world setting, where samples with unknown class labels are continuously observed such that training and testing data may have different class label spaces. Several approaches have been proposed for open-world learning problems in computer vision and natural language processing, but they cannot be applied directly to structured sequences due to the complexity of their non-Euclidean properties and their dynamic nature. This paper addresses this important research gap by proposing a novel Open-world Structured Sequence node Classification (OSSC) model, to learn from structured sequences in an open-world setting. OSSC captures the structural and temporal information via a GCN-based dynamic variational framework. A latent distribution sequence is learned for each node using both stochastic states and deterministic states, to capture the evolution of node attributes and topology, followed by a sampling process to generate node representations. An open-world classification loss is further adopted to ensure that node representations are sensitive to unknown classes. And a combination of Openmax and Softmax is utilized to recognize nodes from unknown classes and to classify others to one of the known classes. Experiments on real-world datasets show that the proposed OSSC method is capable of learning accurate open-world node classifiers from structured sequence data.