RISC-V processors encounter substantial challenges in deploying multi-precision deep neural networks (DNNs) due to their restricted precision support, constrained throughput, and suboptimal dataflow design. To tackle these challenges, a scalable RISC-V vector (RVV) processor, namely SPEED, is proposed to enable efficient multi-precision DNN inference by innovations from customized instructions, hardware architecture, and dataflow mapping. Firstly, dedicated customized RISC-V instructions are proposed based on RVV extensions, providing SPEED with fine-grained control over processing precision ranging from 4 to 16 bits. Secondly, a parameterized multi-precision systolic array unit is incorporated within the scalable module to enhance parallel processing capability and data reuse opportunities. Finally, a mixed multi-precision dataflow strategy, compatible with different convolution kernels and data precision, is proposed to effectively improve data utilization and computational efficiency. We perform synthesis of SPEED in TSMC 28nm technology. The experimental results demonstrate that SPEED achieves a peak throughput of 287.41 GOPS and an energy efficiency of 1335.79 GOPS/W at 4-bit precision condition, respectively. Moreover, when compared to the pioneer open-source vector processor Ara, SPEED provides an area efficiency improvement of 2.04$\times$ and 1.63$\times$ under 16-bit and 8-bit precision conditions, respectively, which shows SPEED's significant potential for efficient multi-precision DNN inference.