Background: The neutron-rich Ba isotopes with only six valence protons represent the beginning of the collective rotation-vibration band structure. The sharp shape phase transition at N=88-90 observed in Ce-Gd isotones is not exhibited in Ba144-146, which renders their analysis interesting. Also there are ambiguities in the spin and band assignments. Purpose: To study their spectra empirically and to compare with predictions from an interacting boson model and microscopic dynamic pairing plus quadrupole model to explain the smooth shape transition at N=88-90. Method: We compare the results of the calculation in the interacting boson models (IBM-1) and IBM-2 and the dynamic pairing plus quadrupole model with experiment and illustrate the variation in level structure of the Ba isotopes with N. The absence of sharp phase transition at N=88-90 is examined. Results: The ambiguous spin and parity of levels of the vibrational bands are assigned on the basis of calculated K components and the decay characteristics. Conclusion: The second Iπ=2+ states in Ba144,146 have K=0 predominance, and Iπ=23+ states are K=2. The smooth transition at N=88-90 is explained.