The Cluster Lensing And Supernovae survey with Hubble (CLASH) is an Hubble Space Telescope (HST) Multi-Cycle Treasury program observing 25 massive galaxy clusters. CLASH observations are carried out in 16 bands from UV to NIR to derive accurate and reliable estimates of photometric redshifts. We present the CLASH photometric redshifts and study the photometric redshift accuracy of the arcs in more detail for the case of MACS1206.2-0847. We use the publicly available Le Phare and BPZ photometric redshift codes on 17 CLASH galaxy clusters. Using Le Phare code for objects with StoN>=10, we reach a precision of 3%(1+z) for the strong lensing arcs, which is reduced to 2.4%(1+z) after removing outliers. For galaxies in the cluster field the corresponding values are 4%(1+z) and 3%(1+z). Using mock galaxy catalogues, we show that 3%(1+z) precision is what one would expect from the CLASH photometry when taking into account extinction from dust, emission lines and the finite range of SEDs included in the photo-z template library. We study photo-z results for different aperture photometry and find that the SExtractor isophotal photometry works best. Le Phare and BPZ give similar photo-z results for the strong lensing arcs as well as galaxies of the cluster field. Results are improved when optimizing the photometric aperture shape showing an optimal aperture size around 1 radius giving results which are equivalent to isophotal photometry. Tailored photometry of the arcs improve the photo-z results.