Concerned About Casters?
There is a persistent current in the D&D audience that runs something like this:
"My god! Wizards! They are OVERPOWERED! TOO MUCH! HOW!? WHY?!"
Personally, just to put my cards on the table, I haven't ever had much of a problem with it, in any game I've played or run in about 15 years. But as religious studies and cultural anthropology (my two degrees) teach you, whenever you see this much consensus on a thing, even if you don't see it, it's always smart to treat it as at least real to the people talking about it, because it's very real to them. At the very least, it's a zeitgeist.
So I accept that there have been issues with illusionists and difficulties with diviners and problems with pyromancers, but I approach it from the outside looking in.
What Is Balance?
To make things very clear, we need to share some sort of definition of what "balance" is. What Good Is Balance? does a pretty good job of it, and so that's what I'm using.
The tl;dr version is that balance exists to create a sense of fairness, and so is basically defined as each character having a roughly equal chance to participate in the game session. A character might suffer through a particular area they're weak in, or champion a particular area they're good in, but each character should go through that cycle in each session, to keep the players interested and not giving up in frustration.
For our purposes here, this means that a player with a wizard contributes to each adventure without overpowering the rest of the characters, and gets to feel both weak and strong.
Balancing Wizards Through the Ages
Early D&D's concept of a wizard was THE powerful arcane spellcaster. They could cast a few powerful spells each day, assuming that they weren't jostled in the casting. These spells would be largely randomly discovered during gameplay, with a chance of not learning them, but could also be researched. Wizards were also weak at 1st level, only becoming stronger in the late-game levels, when they accessed the truly epic kinds of magic that legends are written of.
Modern D&D concieves as the wizard as one of many arcane character types. They still cast spells, though the frequency and power levels are vastly more variable (from a 3e warlock's at-will everything to a 4e swordmage's encounter powers). They are largely considered to have access to their spells, and to use them as their main way of solving problems in an adventure.
There are a few ways in which wizards have been considered unbalanced.
Wizards are considered to be unbalanced at later levels, when they learn magic that eclipses anything a non-spellcaster can do, while the non-spellcasters are stuck using slightly better versions of the same tricks they knew at 1st level.
Wizards are considered unbalanced over the course of an adventuring day, when their spells dominate the resolution of the adventure, and thus they rest whenever their spells are expended, making sure they are always at full power. Characters with more "reliable" attacks and defenses never get a chance to succeed due to the spellcasters always having their most powerful spells available.
Wizards gain spells that can do what any other party member can do, thus rendering any other party member obsolete.
What those methods of unbalancing have in common is that a wizard becomes a force that dominates the game in one way or another. Their spells become omnipotent, their novas become irresistible, and their versatility becomes unmatched. These are absolutely unbalanced results,