The Nintendo DSi has support for WiFi networks, including WPA. Most personal WPA networks use a passphrase instead of the hexadecimal keys used in older WEP setups. The WPA standard supports passphrases of 8 to 63 printable ASCII characters, which are converted to a 64 digit hexadecimal number using a hash of the passphrase and the SSID (see more at WikiPedia).
Unfortunately, the DSi does not allow you to enter a passphrase, you have to enter the 64 digit hexadecimal key. If your router doesn’t show the key (mine doesn’t), you can calculate it using the WPA PSK Generator from the authors of Wireshark (a great network analysis tool).
If you have access to a Linux system with the wpa_supplicant tools installed, you can do this at the command line using:
wpa_passphrase ssid passphrase