We present high-resolution ( \sim 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 \times 10 ^ { 23 } to \sim 10 ^ { 26 } W~ { } Hz ^ { -1 } and radio spectral indices \alpha _ { 1.5 } ^ { 30 } ( defined as the slope between 1.5 and 30 GHz ) distributed from \sim - 1 to -0.25 around the central value \langle \alpha \rangle = -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 ( AGN ) . 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 .