I have two step kids. and I'll give you the pros and con's of them.
DS:
Pro's"
Each kid get's one, no fighting
Rechargeable
Can play both fun games and learning games.
Kids can Sync the DS's and play against eachother from up to 50 feet away (We have been known to be driving in seperate cars and the kids can play the wireless is THAT good on it)
Can Get on the internet with the right software
DS Con's
Twice the amount of games to buy since each child will want specific ones.
very sedentary type games
Cant play full games in the Dual Sync mode
Wii Pro's:
Kids actually have to get active on quite a few of the games
more people can play it at one time
even the parents have games that they can play
Internet availability
Options for games are so much better, ie: Guitar hero, band games, Wii Fit etc.
Compatible with the previous Nintendo Game Cube games. and I believe there is also the option to play the DS games if you buy an adapter.
Wii Con's
Definately not portable.
many different parts are required IE: sports set for the sports game,
Getting to fight the "it's my turn to play!"
The game system and generally the games are more expensive