You've hit the nail on the head with that one. Original Nintendo DS games only support WiFi networks secured with a WEP key, or open WiFi networks.
However the underlying issue here is that there is nobody around with another DS/3DS and Pokemon game for you to trade with. Even if you had a WiFi connection, you could only use it to trade locally with someone else on the same WiFi.
If, somehow, you were able to get your hands on another DS/3DS and another game of the same generation, then you could use it to trade locally via WiFi. Almost all (if not all) consumer routers support opening the WiFi network by disabling security altogether. Consult your router's documentation or online resources (using the router's model number) on how to do this. The simplest solution is to back up your existing security information, setting it as an open network for the trade, then setting the security back onto the router.