Efficient quantum state measurement is important for maximizing the extracted information from a quantum system. For multi-qubit quantum processors in particular, the development of a scalable architecture for rapid and high-fidelity readout remains a critical unresolved problem. Here we propose reservoir computing as a resource-efficient solution to quantum state readout of superconducting multi-qubit systems. We consider a small network of Kerr oscillators realized by Josephson parametric oscillators, which can be implemented with minimal device overhead and in the same platform as the measured quantum system. We theoretically analyze the operation of this Kerr network as a reservoir computer to classify stochastic time-dependent signals subject to quantum statistical features. We apply this reservoir computer to the task of multinomial classification of measurement trajectories from joint multi-qubit readout. For a two-qubit dispersive measurement under realistic conditions we demonstrate a classification fidelity reliably exceeding that of an optimal linear filter, while simultaneously requiring far less calibration data. These results are obtained for reservoirs of two to five nodes trained with as few as 10 samples per state. We understand this remarkable performance through an analysis of the network dynamics and develop an intuitive picture of reservoir processing generally. This reservoir-classifier avoids computationally intensive training common to other deep learning frameworks and can be implemented as an integrated cryogenic superconducting device for low-latency processing of quantum signals on the computational edge.