It is a wrong statement because constant as well as variables can be dimensional as well as non dimensional.
Constants like 1,2,3,4,5, , e, $\pi $, sin $\theta $, cos $\theta $, tan $\theta $ etc., are dimensionless constants but constants like speed of light in vacuum c, Planck's constant h, Gravitational constant etc. are dimensional constant. Variable like volume, force, velocity, frequency etc. have dimensions but variables like strain, angle, specific gravity etc., are dimensionless