In the downlink of the Cloud-Radio Access Network (C-RAN) cellular architecture, complex baseband signals are transmitted from a central unit (CU) in the cloud over digital fronthaul links to distributed radio units (RUs). The standard design of digital fronthauling is based on quantization that operates separately over each fronthaul link. In this paper, a fronthaul quantization scheme is proposed that, unlike conventional schemes, implements a joint quantization mapping across all fronthaul links that is adapted to the current channel conditions. As compared to the current standard approach, the proposed multivariate quantization (MQ) scheme only requires additional processing at the CU, while no modification is needed at the RUs. The algorithm is extended to enable variable-length compression, and is compared via numerical results to a related approach based on the information-theoretic technique of multivariate compression.