Best bet is prot. People *always* needs tanks. People rarely need bad* dps that can't cc though.
*gets better, but only at t5/t6 level.
Agreed, our guild has Karazhan on farm status and the only melee DPS class that we take with us is Rogue and even then we only allow one Rogue to come with us. Basically the game's instances are best suited for ranged DPS and thus you will almost always have Mages, Warlocks, Hunters, Shadow Priests, Elemental Shamans and Moonkin druids picked before you.
Unfortunately in fights like Maiden and Prince, melee DPS is useless since they're usually the first to die because the healers will be priorotising their heals on tanks. Basically this means that your chances of dying early during a boss fight are very high, thus turning it into a 9 man raid. In the case of ranged DPS, they're far away from the damage and thus the healers rarely have to waste mana on them at all.
The limitations of melee DPS are not only related to Karazhan, any group that wants to increase their chances of success will always take ranged DPS over melee DPS for just about any instance in the game. As long as ranged DPS watches their aggro, the healer will be able to focus healing on the tank only. Add to that, just about all ranged DPS has one or more crowd control abilities, basically means that DPS warriors, Ret Paladins and Cat druids are the bottom of the choice list when they're trying to fill the spots for DPS in any group.
As kHayne suggested, you will probably wish to go tank if you want to increase your chances of getting into heroics and raids. Tanks are always in short supply. I have a level 70 epic geared warlock and a level 70 blue/epic geared tank and I have way more invites to go to instances and raids with my tank. Theres way too much DPS classes out there, thus tanks and healers will always have the biggest demand.
Just remember, if you're going prot, other classes can get by on mediocre gear, tanks on the other hand need to get the best gear as fast as possible. You've probably figured out by now that when the tank dies, theres a 90% chance that the rest of the group is going to be dying next. Do not start heroics or even consider entering karazhan until you have gotten rid of all your green items and ensure that any blue items that you do have are level 70 at minimum.