How can I tell a Windows 2003 R2 server, which has multiple NICs, "listen for DHCP requests on NIC #1, ignore DHCP requests on NIC#2" ? The server is in a gateway role, connecting two networks, and I don't understand how to get it to be a DHCP server only for the "downstream" network.
Presumably you only have a scope setup for your internal network. Your DHCP server will not respond to requests received on an interface that doesn't have an address that is within one of the defined scopes.
So if you have 2 interfaces with addresses 10.23.5.1 and 192.168.32.12, and one scope defined for 192.168.32.0/24. Your DHCP server will only offer 192.168.32.0/24 addresses to hosts when it sees the requests coming in on the 192.168.32.12 interface.
It doesn't sound like it applies in your case, but to extend the answer a little more. If you have DHCP relay agents setup they will include the address of the interface the request is received on as part of the forwarded DHCP request. The DHCP server will use that forwarded address from the agent to decide what the correct scope is to serve an address from.