Hmm...
The 'core four'--fighter, mage, cleric, rogue--are essential, of course. I'd quite happily see a D&D without the cleric, but that chance was missed twenty years ago; at this point, there's just too much legacy.
Ranger and paladin follow close behind. The paladin is arguably a prestige class/paragon path/fighter-cleric template, but is an archetype I want in the game, and the ranger is nearly as core as the basic four. I might, however, give the paladin some of the cleric's 'warrior-priest' element and shift the cleric more towards a White Mage/4E invoker archetype.
Maybe my MERP and 2E influences are showing, but I'd like the bard in there. 'Magical music' would be only one of its potential paths, although with the 'people person' and 'jack of all trades' elements.
From 4E, I'd take the warlord (possibly renamed 'captain' or something that sounds less elite), for the 'non-magical healing' option it provides and because it fills another valuable fantasy archetype--the heroic 'leader of men'. Perhaps blend elements of the cavalier with it for a broader 'noble warrior' archetype.
Druid, monk and barbarian belong to the expansion material. Those classes have always had a flavor of the exotic, and thus should be left a bit more out of the core.
Warlock? Most decidedly not in the core. Any D&D I'm in charge of is going to have a bright shiny Epic/High Fantasy feel to it, and while there's room for the 'evil-curious' hero, it belongs in an expansion, not as a core class.
The sorcerer and illusionist both belong in a magic expansion book, along with the other specialist mages. The swordmage or something like it, however, merits core status to fill the 'warrior mage' niche.
So, we wind up with:
Fighter
Ranger
Paladin (possibly with a bit more magical oomph than previously--probably something that looks a lot like the 4E paladin, really)
'Captain'/Warlord
Mage
Swordmage/'Warrior Mage'
Cleric
Rogue
Bard