The evolution of Internet and its related communication technologies have consistently increased the risk of cyber-attacks. In this context, a crucial role is played by Intrusion Detection Systems (IDSs), which are security devices designed to identify and mitigate attacks to modern networks. Data-driven approaches based on Machine Learning (ML) have gained more and more popularity for executing the classification tasks required by signature-based IDSs. However, typical ML models adopted for this purpose do not properly take into account the uncertainty associated with their prediction. This poses significant challenges, as they tend to produce misleadingly high classification scores for both misclassified inputs and inputs belonging to unknown classes (e.g. novel attacks), limiting the trustworthiness of existing ML-based solutions. In this paper, we argue that ML-based IDSs should always provide accurate uncertainty quantification to avoid overconfident predictions. In fact, an uncertainty-aware classification would be beneficial to enhance closed-set classification performance, would make it possible to carry out Active Learning, and would help recognize inputs of unknown classes as truly unknowns, unlocking open-set classification capabilities and Out-of-Distribution (OoD) detection. To verify it, we compare various ML-based methods for uncertainty quantification and for open-set classification, either specifically designed for or tailored to the domain of network intrusion detection. Moreover, we develop a custom model based on Bayesian Neural Networks to ensure reliable uncertainty estimates and improve the OoD detection capabilities, thus showing how proper uncertainty quantification can be exploited to significantly enhance the trustworthiness of ML-based IDSs.