What characteristics of magentic coil is the most important for VLF sensitivity
Generally speaking, the "loop area" (the area inside the coils of wire) is the primary factor that determines the sensitivity of a loop antenna.

If the noise is also magnetic in nature, there isn't much you can do other than try different orientations of the coil to maximize the signal-to-noise ratio. Increasing the loop area in this case will only increase the noise, too.

However, a large coil can also couple noise from electric fields to the circuitry it's driving, and it may therefore be helpful to wrap your coil in a grounded electrostatic shield. I've seen loop antennas created by pushing multiconductor cable into a piece of copper tubing, bending the tubing into a circle, and then connecting all of the conductors in series to create a coil. But be careful to leave a gap in any such shield; otherwise, it will create a "shorted turn" and eliminate any possibility of detecting magnetic fields.