Such a blanket ban seems a bit severe, and as others pointed out, it won't necessarily prevent the abuses that can occur with just core.
Myself, I'd probably start by banning non-core base classes. I feel they tend to contribute in part to the whole "Tier" problem that plagues 3.5, though some of those classes (not which ones specifically, though it really doesn't matter to me) I know are considered decidedly subpar. I think it doesn't help when there are classes that can, say fight as well as a fighter but bring other abilties to the table, because they make the fighter even weaker. Still, simply banning base classes won't fix this problem; core still has paladins and rangers (not to mention melee focused clerics or druids) which can outshine the fighter, and rogues can be outclassed by a decent bard build or the right wizard spells. And monk has no real focus to its abilities. Then again, I'm not really fond of the 3.5 base classes anyway; I don't like the flavor, and I'd prefer to see character customization achieved through feat choices and multiclassing rather than whole new classes. And even getting rid of non-core base classes doesn't knock a single class out of Tier 1, which is where a lot of the worst abuse can occur, and without even leaving core.
Multiclassing has problems when people cherry pick abilties from the first few levels of a class for the benefits. It's not just a power problem, it just makes the character seem artificial rather than organic. I wouldn't want to dump multiclassing entirely, because as I stated above, this is how I like to see characters customized. I just prefer to see it done in a non-abuse fashion. I'm not opposed to prestige class use, but it should be kept firmly under the DM's control.
Spells are another thing the DM needs to keep control over. Wizard spells are the easiest to control, since they're restricted to what's in the wizard's spellbook. The main thing to restrict here are which spells the wizard starts with and which spells are learned each level. I typically restrict these to PHB spells only. Anything else the wizard needs to find a copy of the spell in a book or scroll, and I control which spells become available that way. The bigger issue is with clerics and druids since they don't have a limiting factor like a spellbook. I would at least put some feature in place where they actually have to earn the ability to cast a spell that's not in the PHB, if I want to allow it.