Wireless positioning systems that are implemented by means of a Cloud Radio Access Networks (C-RANs) may provide cost-effective solutions, particularly for indoor localization. In a C-RAN, the baseband processing, including localization, is carried out at a centralized control unit (CU) based on quantized baseband signals received from the RUs over finite-capacity fronthaul links. In this paper, the problem of maximizing the localization accuracy over fronthaul quantization/compression is formulated by adopting the Cramér-Rao bound (CRB) on the localization accuracy as the performance metric of interest and information-theoretic bounds on the compression rate. The analysis explicitly accounts for the uncertainty of parameters at the CU via a robust, or worst-case, optimization formulation. The proposed algorithm leverages the Charnes-Cooper transformation and Difference-of-Convex (DC) programming, and is validated via numerical results.