Hi,
the diameter of the screw determines the stiffness/rigidity of it, obviously the larger the better from the point of view of strength.
A more subtle effective minimum diameter is the maximum rotational speed.
The most comprehensive deign guide I have seen on the subject comes from the NSK Precision Components catalogue:
http://www.nsk.com/common/data/ctrgPdf/e3162h.pdf
Section B-2-3
The important parameters are the max unsupported length, that is the max distance between the ball nut and either of the support units, and the nature
of the support units, ie fixed, floating etc.