You can make it work, but you need to go into the router settings and make sure that backwards compatibility for wireless "b" is enabled. Some newer routers disable it for performance reasons. Then you need to change the authentication + encryption to WPA-PSK and AES (or maybe TKIP). I think you'll then need to re-connect all your existing devices since the authentication method changed. It's a huge pain in the ass. I'm not sure it's worth it for a PSP.
If your new router has dual 2.4GHz and 5GHz networks (most new routers do), then you can make these changes to just the 2.4GHz side and leave the 5GHz alone. You'll need to make sure each side has a different SSID though.
It might also be possible to set up a second older router with a different SSID just for the PSP but I'm not sure about the specifics.
GroovyMAME support forum on BYOAC