Here is your fix... and it works.
I spent an hour on the phone with Nintendo, and we got this to work AND keep a secure network at the same time.
Apple is pushing a new type of encryption key that is a 13 digit code. The Nintendo DS will not support a 13 digit code. Only 8, 10 and 26 digit codes in WEP format. The WEP format that Apple uses on the new Airport Extreme N routers is different. If you read what it says in the description on the WEP setting is says "A Transitional Security Network (TSN) provides security that is compatible with older WEP clients, but allows newer clients to join the network using WPA and WPA2. You must enter a password of exactly 13 characters."
That is the problem with the DS and why you get a 51300 error when you try to join an encrypted network with and Airport Extreme Router. The DS does not support the 13 key, only 10. So, here is the fix, and how to remain a secure network.
Open up your Airport Utility. Click on Manual Setup, then click on the Wireless button in the bar across the top middle of the window. Leave the SSID or Network Name alone, select your Radio Mode to what ever you want. any of them will work with DS. Change it to Channel 11 and Wireless Security, change to NONE.
Second step: Click on Access Control next to the Wireless button you pressed to get to the Wireless settings. Then under MAC Address Access Control, select Timed Access. Double click on one that says "Default" and so it opens and change it to NO ACCESS. This will close your network. Now, get the MAC address, of your DS (Which is found in the WIFI Settings) and enter it in using the + at the bottom of the MAC Address Access Control section. If you want it on all the time, then change it to Everyday and All Day so it dont expire. You will also need to do this for any computer, DS or any wireless device that connects WIRELESSLY to your network. If they are hardwired into the Airport Extreme, then you don't have to worry about it.
When your done, click "UPDATE" and let the router reset. Now fire up your DS and Voila! Your gamin online.
Hope this helps, it took me months to figure it out and I scoured the web looking for an answer. I also asked Nintendo to post it on their website.
~ Joe