We present high-resolution (similar to 1 ''), 1.5 GHz continuum observations of the brightest cluster galaxies (BCGs) of 13 CLASH (Cluster Lensing And Supernova survey with Hubble) clusters at 0.18 < z < 0.69 with the Karl G. Jansky Very Large Array (JVLA). Radio emission is clearly detected and characterized for 11 BCGs, while for two of them we obtain only upper limits to their radio flux (<0.1 mJy at 5 sigma confidence level). We also consider five additional clusters whose BCG is detected in FIRST or NVSS. We find radio powers in the range from 2 x 10(23) to similar to 10(26) W Hz(-1) and radio spectral indices alpha(30)(1.5) (defined as the slope between 1.5 and 30 GHz) distributed from similar to-1 to -0.25 around the central value = -0.68. The radio emission from the BCGs is resolved in three cases (Abell 383, MACS J1931, and RX J2129), and unresolved or marginally resolved in the remaining eight cases observed with JVLA. In all the cases the BCGs are consistent with being powered by active galactic nuclei. The radio power shows a positive correlation with the BCG star formation rate, and a negative correlation with the central entropy of the surrounding intracluster medium (ICM) except in two cases (MACS J1206 and CL J1226). Finally, over the restricted range in radio power sampled by the CLASH BCGs, we observe a significant scatter between the radio power and the average mechanical power stored in the ICM cavities.