The Cloud-Radio Access Network (C-RAN) cellular architecture relies on the transfer of complex baseband signals to and from a central unit (CU) over digital fronthaul links to enable the virtualization of the baseband processing functionalities of distributed radio units (RUs). The standard design of digital fronthauling is based on either scalar quantization or on more sophisticated point to-point compression techniques operating on baseband signals. Motivated by network-information theoretic results, techniques for fronthaul quantization and compression that improve over point-to-point solutions by allowing for joint processing across multiple fronthaul links at the CU have been recently proposed for both the uplink and the downlink. For the downlink, a form of joint compression, known in network information theory as multivariate compression, was shown to be advantageous under a non-constructive asymptotic information-theoretic framework. In this paper, instead, the design of a practical symbol-by-symbol fronthaul quantization algorithm that implements the idea of multivariate compression is investigated for the C-RAN downlink. As compared to current standards, the proposed multivariate quantization (MQ) only requires changes in the CU processing while no modification is needed at the RUs. The algorithm is extended to enable the joint optimization of downlink precoding and quantization, reduced-complexity MQ via successive block quantization, and variable-length compression. Numerical results, which include performance evaluations over standard cellular models, demonstrate the advantages of MQ and the merits of a joint optimization with precoding.