Long story short, the patchers are no longer needed. They worked on any flash card, so your specific model doesn't matter, but what they did was force games to connect over HTTP instead of HTTPS, but it has since been discovered that the DS HTTPS implementation is super busted, so now you can just connect to servers with stock cartridges/ROMs.
There are several WFC replacement servers, but two main ones: the open-source AltWFC
(which there are several servers for, including the official one) and closed-source Wiimmfi
(of which there is only one). These operate in much the same way, probably the most visible difference to end-users is that AltWFC will accept connections from any game you try to connect with, while Wiimmfi focuses on working with a specific list of games. Personally, I would strongly recommend AltWFC, but you do whatever makes you happy (or use whichever one your friends are on). You can connect to these servers by changing the DNS server on your 3DS; the web sites should have instructions on how to do this.
The other thing to consider is that DS games only support WEP or no encryption on your wi-fi, whereas the Wii, DSi, 3DS and probably everything else you own supports WPA. This means even when running those DS games on a newer system like a 3DS, you're still limited to WEP, since that's what the games use. If you want to play DS games online, you'll need to set up a WEP/unencrypted access point on your router (preferably) or on your phone (if necessary). Lots of routers allow you to set up a separate "Guest Network" which you could set up with WEP if you want, instead of messing with your actual household Internet access point.
A quick aside: WEP is supremely unsecure, it's very
easy for literally anybody, with zero technical know-how, to crack your WEP password in minutes, so running an access point with WEP does potentially invite people to steal your Internet and do who knows what with it. This is another reason to keep it as a guest network, since a) it walls the network off from your other devices which is nice for security and b) you can turn the guest network on and off at will, i.e. only enable it when you're using it. Depending on your router, you might also be able to use MAC filtering so that only your DS is allowed to connect to the guest network, but that's easy to bypass too--you're not going to come up with a way to make a WEP connection secure, it's ancient and bad and that's just something you have to accept if you want to use it.
On the upside, you can play DSi-enhanced cartridges (not ROMs) and DSiWare via WPA encrypted wifi using a replacement server, since the DSi was not limited to WEP only.