Sorry about your problem.
Before you look at this and think “Good Lord that’s a complicated solution.” I want you to know that I went into as much detail about setting up the Time Capsule as possible because I’m not sure how familiar you are with Macs. It should be a very descriptive step-by-step tutorial on how to get your DS working on Nintendo Wi-Fi Connection again.
I know how to fix it though. On the Mac, click Finder (the left-most icon on the dock) choose Applications. Scroll down and open Utilities. Double click “AirPort Utility.app” to open the application. It should pull up your Time Capsule and it should be running fine as it is…but it won’t let your DS connect. There is ONE way that I’ve found (through MUCH trial and error) to let your DS connect to it and still maintain a pretty secure network. Click “Manual Setup” with the Time Capsule selected in AirPort Utility. Select the “Wireless” tab towards the top of the window. For “Wireless Security” select “None.” I know this sounds risky, but trust me. Once you have no wireless security on your network, click “Wireless Options…” Check the box for “Create a closed network.” What this does is make your network hidden. People can only connect to it if they know the name of your network and actually tell their device to look for a network named whatever your network is. To save these settings, click “Done” and then “Update.” Before you click “Update” though, you may want to come up with a pretty unique Wireless Network Name. Type in something you could easily remember but no one would ever guess. Mine is the first letter of every member of my household. Like, if my name were Jack, my wife’s name Daisy, son’s name Greg, and daughter’s name Jill, my network name might be jdgj. Easy to remember, since it’s my family in order according to age, and no one is going to run a search for a network named that. You’ll be completely safe. If people want to connect to someone else’s wireless, they’ll go for the unsecured ones that are visible just by searching, or try to guess the password to the visible ones that are secured.
Now, for your DS… I don’t have my DS with me right now, so I’ll do my best to explain how to do it. Go to your Wireless Settings via a Nintendo Wi-Fi Connection enabled game and pick a connection (preferably Connection 1 as that’s the first one your system looks for). It may or may not run a search automatically. What you want to do is manually type in the ID of your network, which is whatever you set in AirPort Utility. You have to manually type it in since it’s a closed network. I believe after that all you have to do is tell it to try to connect to a network with that Wireless Network Name and it’ll set the rest up in a jiffy. If you have any problems, leave another post here and I’ll whip out my old DS Phat and get more detailed about that portion. Hope it helps!