Internet browser (as HTTP) shouldn't require any open ports.
Opening ports is used to allow incoming request (when you host a http/ftp/game server for example).
But accessing the net, the port 80 should be granted automatically.
80 for normal connections, and 443 for secure connections.
For online games though, you have to open some ports.
I've checked Nintendo's website and that's right, they are telling the users to open
ALL ports, from 1 to 65535, AND redirect them all to the 3DS

the computer won't work correctly anymore if you do that
You could start by opening only the ports used by other online games:
(these ports are used by wii and DS games, but it could be always the same for all their 3DS games too).
TCP: 28910, 29900, 29901, 29920
TCP: 80, 443 (Shouldn't be useful to open them, but they are recommending it. Don't open them if not needed).
You can redirect them to the IP of your 3DS. (set the 3DS as fixed IP).
They are also recommending to open/redirect ALL UDP ports to 3DS (don't do that either. opening/redirecting only the used ports is better).