Before 3E and other "modern" editions of D&D, wizards were definitely more powerful than fighters at high levels.
My experiences are mostly from AD&D 2nd Edition. At low levels (roughly 1-5), wizards needed fighters to keep them alive. While they knew a trick or two, low-level wizards didn't really get anything done in a fight. The game was most balanced with mid-level (level 5-10) characters, when the classes complemented each other quite nicely. Fighters held the line and enganged enemies that dealt heavy damage, while wizards controlled the battlefield, wiped out masses of low-level enemies, and dealt heavy damage themselves. Things changed beyond level 10. While fighters didn't really grow any more powerful, wizards could cast more spells and got access to even more powerful ones.
3E changed the balance in two ways. First, the introduction of feats, prestige classes, and other special effects meant that high-level fighters continued gaining new abilities and dealt more damage. Second, the proliferation of magic items added even more abilities to non-spellcasters.
To some degree, this all depended on the playstyle of the group. While fighters were reactive characters, wizards were active characters. If the group regularly ended up in an ambush at close quarters, fighters were the ones who would get them out of trouble. On the other hand, if the group preferred to ambush their enemies and fought over long distances, wizards were the ones who did the real fighting. To put it another way, while a high-level fighter could arguably beat a high-level wizard in a fair fight, the wizards who had a habit of ending up in a fair fight with fighters already died at low levels.