This is the common impression.
But, is it just a rule of thumb and correct only in some wavelength region or does it hold universally?
It is more important in the sense of accelerating charged particles.
In vacuum, it is easy to show that the ratio of the amplitudes of the electric field and magnetic field is $c$ (in SI units).
The Lorentz force on a charged particle is $$\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})$$
If we just consider the magnitudes of the electric versus the magnetic forces acting on a charged particle, then (for a charge in vacuum), we can see that $$ \frac{F_B}{F_E} = \frac{v}{c}$$
Thus unless the charges are moving relativistically then it is the electric part of the Lorentz force that is dominant.
In a conducting material, the balance between electric and magnetic fields changes. In a very good conductor then then $$\frac{E}{B} = \left( \frac{\omega}{\mu_0 \mu_r \sigma}\right)^{1/2} = c\left( \frac{\omega \epsilon_0}{\sigma}\right)^{1/2},$$ with a a phase difference of $\pi/4$ between the E- and B-fields. In general this ratio will be $\ll c$ in a good conductor where $\sigma \gg \omega \epsilon_0$.