How do I set influence radius to a specific size? When doing proportional edits in Blender is there a way to set the radius of influence to a specific size (e.g. 23) or do I have to use my mouse wheel and just try to get close?
As with most operations in blender, the redo panel (F9 in 2.8+ keymap, or  F6 for 2.7 keymap) provides the ability to adjust the operator immediately after it is run. Keep in mind, if you use another operator (scale, translate, etc before adjusting prop edit), then the ability to adjust is gone.
See the gif below: