Like has been mentioned by some already; if you are starting the campaign at a higher level, then have it start there. There is no rule that says that a campaign has to start at level 1. There are a couple of different ways you can handle this.
Existing Party: Like mxyzplk suggests. The group is an existing party that has already adventured together, but their last adventure has ended. The villain has been defeated, the princess saved, and the loose ends tied up.
And then another adventure starts. Maybe the king has another unrelated quest. Maybe their reputations have spread, and they are sought out by a grand wizard. Maybe a new evil rears its head, and the adventurers seek it out on their own accord.
Benefits: The group has a good reason to travel together. They can draw on past contacts (which you can make up on the spot "Oh, you need help identifying the potion? There was that alchemist you saved during your last adventure.")
Drawbacks: You (GM and players) either need to make a solid background specifying what that past adventure was about and how it went, or you need to accept that there is a good chunk of background that isn't really specified. The first is potentially a lot of work, the other can lead to a lot of questions ("Do we know the mayor of this town? Have we even been here before?") This can be alleviated by setting the new adventure in a distant land from the first one.
Existing heroes: The heroes are already established badasses, but haven't adventured together before. One might be a wizard with his own tower. One might be a knight of the realm. One might be the best cat burglar in all of Asharn. They then end up together in an adventure due to circumstances, or they are called together by an outside force. Just like if they were level 1 adventurers.
Benefits: The players get to be established badasses and have fun with their backgrounds. Who doesn't want to start the adventure as the best cat burglar in all of Asharn? And it is a classic in fiction; a monarch or other authority figure calling together the best people his realm can offer for a mission.
Drawbacks: The group had less of a reason to stick together. This is also true with 1st level groups, but when the wizard has his own tower, and the barbarian is the chieftain of a small tribe, there is greater chance of friction and "I don't need the rest of you"ism. Make sure that players collaborate to make a group that can work together. Friction between the paladin of the realm and the realm's best assassin can be fun, as long as it remains differences of opinion and not party-splitting quarrels.
New heroes: Maybe the heroes just start off at level 10. Maybe there are no level 1-9 wizards and fighters. Once your apprenticeship is done, you are level 10 with all the spells and feats that this encompasses. This is likely to be an ultra-high fantasy campaign with continual flame on every street corner, and wards on every door and window to keep out the flying, knock-equipped rogues.
Benefits: An ultra-high fantasy campaign might be a fun jaunt. In this case, the benefit is probably all about whether you like the setting or not.
Drawbacks: All the opponents and NPCs should probably be 10+ too. Have fun stating that up.
Empowered heroes: Maybe the heroes just start off at level 10. They were experimented on by an evil mage. Or wandered into the fairy forest. Or touched the Artifact of Divine Heroism. And now they have all these powers. And their first quest is probably tied into whatever it was that empowered them. They probably start off without any magic items, but as long as you keep this in mind when picking out the monsters, and let magic items drop at a steady pace until they are geared up, it shouldn't be a problem.
Benefits: They are just like level 1 characters, except they are level 10. And whatever empowered them is likely to both tie the group together, and provide a good plothook.
Drawbacks: Some weirdness might arise (the Leadership feat works how?). Might feel more like a super hero starting story than a fantasy starting story (though some might consider that a benefit). Be careful that the starting hook isn't too constricting, unless you know your players will be okay with that.