I am not an optics expert, but since light inside an optical fiber is contained by reflection, the only limit is the heat dissipation of the fiber and surrounding materials. Perhaps someone smarter than I can comment on how diffraction comes into play at the higher intensities.
A good baseline would be to look at metal cutting cnc lasers and what diameter of fiber optic line they are using. The one in the link is 2,000 W at (just guessing) about 6 mm diameter fiber. This is for 1064 nm, but most of the solar spectrum that is transmissible through glass is at a shorter wavelength so you would be able do do 2,000 W or better. Sunlight is roughly 1,000 W per square meter.
The economic constraint to this however is your ability to focus that much sunlight into the small diameter cable and at an angle reasonably perpendicular to the polished face of the optical fiber. This optical fiber sun tracking system looks like a good solution. So ultimately the diameter of fiber you choose will be determined based on your fiber cost per foot, focusing system cost, and depth of bunker.
Often well drilling systems have a minimum diameter on the order of 6". If you are already drilling/digging a vertical hole to your bunker, putting in a 6" diameter mirrored light tube would be a much easier and lower cost solution. If you still need to keep the nukes, zombies, and radiation out just use some extra thick glass at both ends ;-)