Context. Basaltic asteroids (spectrally classified as V-types) are believed to be fragments of large differentiated bodies. The majority of them are found in the inner part of the asteroid belt, and are current or past members of the Vesta family. Recently, some V-type asteroids have been discovered far from the Vesta family supporting the hypothesis of the presence of multiple basaltic asteroids in the early solar system. The discovery of basaltic asteroids in the outer belt challenged the models of the radial extent and the variability of the temperature distribution in the early solar system. Aims: We aim to identify new basaltic V-type asteroids using near-infrared colors of 40 000 asteroids observed by the VHS-VISTA survey and compiled in the MOVIS-C catalogue. We also want to study their near-infrared colors and to study the near-infrared color distribution of the Vesta dynamical family. Methods: We performed a search in the MOVIS-C catalogue of all the asteroids with (Y-J) and (J-Ks) in the range (Y-J) ≥ 0.5 and (J-Ks) ≤ 0.3, associated with V-type asteroids, and studied their color distribution. We have also analyzed the near-infrared color distribution of 273 asteroid members of the Vesta family and compared them with the albedo and visible colors from WISE and SDSS data. We determined the fraction of V-type asteroids in the family. Results: We found 477 V-type candidates in MOVIS-C, 244 of them outside the Vesta dynamical family. We identified 19 V-type asteroids beyond the 3:1 mean motion resonance, 6 of them in the outer main belt, and 16 V-types in the inner main belt with proper inclination ip ≤ 3.0°, well below the inclination of the Vesta family. We computed that 85% of the members of the Vesta dynamical family are V-type asteroids, and only 1-2% are primitive class asteroids and unlikely members of the family. Conclusions: This work almost doubles the sample of basaltic asteroid candidates in regions outside the Vesta family. Spectroscopic studies in the near-infrared and dynamical studies are needed to confirm their basaltic composition and to determine their origin.