I run from enemies that give so little EXP that it's not worth it (loads of wolves), enemies that can poison you, enemies that can petrify you, enemies that can instantly paralyze and kill you (Sorcerers), enemies that are just plain deadly (Mages), enemies that are hard to kill (some Ooze enemies in the Marsh Cave, IIRC) and in situations when I need to save magic or make sure my party can survive. If I have a KO'd party member, I don't want him/her to fall behind in EXP, so I just keep running away from battles until I reach a Clinic/Church to revive that character.

In addition to those, I run away from most of the battles in Marsh Cave, Earth Cave and Ice Cave, and even after I get ProRings, I run away from some of the battles in the Castle of Ordeals and the Sea Shrine. During my four games of FF1 WSC, my parties of Fi/Fi/Rm/Wm, Fi/Bb/Wm/Bm, Fi/Th/Rm/Wm and Fi/Rm/Bb/Bm, only the Fi/Th/Rm/Wm party did a good job of escaping battles, and the other parties kept getting annihilated a few or several times in the Ice Cave. Based on that, I wouldn't call the Thief or running away from battles useless.