I use my A4000 for surfing the net via a USB-Ethernet device plugged into a Deneb.
What software are you using to connect? I installed the Internet package from the OS3.9 CD and configured Genesis as follows using teh Wizard:
Select "Network" for how you plan to connect.
For device enter "usbasixeth.device" and set unit as "0"
Under SANA II configuration select "no"
Don't worry about it not detecting your network address, just manually enter your own (my home network assigns 192.168.0.xxx so I used 192.168.0.110).
Enter your subnet mask (probably 255.255.255.0) and your router address (in my case 192.168.0.1)
Enter your Domain Name Server (DNS) - 192.168.0.1 in my case
Enter your Domain Name ("cablerocket.net" in my case).
Reboot your Amiga and check your Trident prefs and you should see your ethernet adapter up and running. Check Genesis and put the USBASIXETH0 device online if it isn't already. Open your web browser and start surfing!
I prefer iBrowse, but I haven't got around to reinstalling it yet.
PS. I got my settings by attaching a PC to my router, opening a DOS window and typing "IPCONFIG /ALL"