How are you connected to the internet, straight to a modem or by a router ?
Can be that you need to set a static IP address rather then a dhcp server.
Check those settings on the other computer, and if you are using a router you have to look at those settings it can also be that you have mac filtering on and your broadcom adapter is not listed there.
As far i can see from those screens, your adapter seems to be running fine.
It just cannot obtain any ip address and is assigned a loop back ip address (127.0.0.1)