Like Zach said above, your router is probably using a WPA encryption, which original DS games cannot access (such as Diamond, Pearl, Platinum, HeartGold, SoulSilver, etc.)
You're going to need to switch the security settings on your router to WEP (from WPA, which is probably what you have now). You'll also probably need to disable your router's firewall. This is exactly what I had to do so I could connect to Wifi on my Diamond version. Honestly, though, the process is pretty lengthy and can be confusing to any average user (like me). What I did was call Nintendo so they could guide me through the process and it took about 20 minutes or so for me to be able to connect. I suggest doing that if you want to be sure you're doing everything right.
If not, I found this guide online, but I haven't tried this or anything, so I still suggest calling Nintendo about the problem, and if you do decide to call them, tell them that you cannot connect to Wifi on your DS because your router is WPA encrypted. They'll be more than happy to help. Regardless, here are the instructions I found:
Accessing your router:
1) Go to your internet browser and type in the IP address of your router
2) The login info is always different between routers. If you’ve never accessed it before it could be:
password: could be “password” or “admin”
3) Go the wireless section then wireless security.
4) You should be able to change from WPA to WEP
Good luck, and hopefully all goes well.