My 360 is hooked up with a ethernet cord. But doesn't the wii/ds wireless adapter plug into a USB port?
The wireless adapter from Nintendo plugs into a USB port on a computer and generates a soft access point, designed for the DS and Wii to connect to wirelessly. However, if you've got a wireless router, you can just have them connect to that instead.
If you want to plug the Wii into a network via Ethernet, you will need the LAN adapter. If you did that, then you could still get one of the USB wireless adapters and use it solely for the DS.
I guess it really comes down to how much you want to get online with both the DS and the Wii versus just the Wii. The adapters are going to be about the same cost either way.