Echo state networks (ESNs), belonging to the wider family of reservoir computing methods, are a powerful tool for the analysis of dynamic data. In an ESN, the input signal is fed to a fixed (possibly large) pool of interconnected neurons, whose state is then read by an adaptable layer to provide the output. This last layer is generally trained via a regularized linear least-squares procedure. In this paper, we consider the more complex problem of training an ESN for classification problems in a semi-supervised setting, wherein only a part of the input sequences are effectively labeled with the desired response. To solve the problem, we combine the standard ESN with a semi-supervised support vector machine ((SVM)-V-3) for training its adaptable connections. Additionally, we propose a novel algorithm for solving the resulting non-convex optimization problem, hinging on a series of successive approximations of the original problem. The resulting procedure is highly customizable and also admits a principled way of parallelizing training over multiple processors/computers. An extensive set of experimental evaluations on audio classification tasks supports the presented semi-supervised ESN as a practical tool for dynamic problems requiring the analysis of partially labeled data.