Ethernet is a highway, over which can travel many types of data packets. The most common is Internet Protocol, (IP) or more formally Transmission Control Protocol/ Internet Protocol (TCP/IP). to work properly, you need to have TCP/IP active, simply because the software assumes that is what is using the Highway. Along that same Highway, you will also be sending AppleTalk packets, which do not interfere with any TCP/IP packets, but do not interact with them, either.
On your PowerBook, I am taking your word for it that you do NOT have the option board that provides an RJ-11 modular telephone jack and an RJ-45 8-wire Ethernet jack. We would not expect your Mac to offer Ethernet as an option in the AppleTalk Control panel. If your Mac offers Ethernet, you need to look more carefully at whether you do indeed have the option card installed.
At the PowerBook, in the AppleTalk Control Panel, you expect to turn on AppleTalk/LocalTalk using the serial Port. I do not remember the exact wording. When you close the Control Panel, it should become active, and a light should come on at the AsantéTalk. You should also use the Sharing or File Sharing Control Panel to turn on File Sharing.
This article has some help with set-up:
http://www.atpm.com/network/setup/localtalk_ethernet.htm
At your 10.3.9 Mac, you should use the System Preferences > Network to look at all your Network Interfaces, select "Built-in Ethernet" and in the TCP/IP pane, give it an IP address Manually if it does not already get one automatically via DHCP. Then in the AppleTalk pane at the right, check "Make AppleTalk Active". Save changes.
On your 10.3.9 Mac, you enable Sharing using System Preferences > Sharing > ("Services" pane) check "Personal File Sharing". Save changes.
To initiate a connection, open the Chooser on the PowerBook, and select AppleShare as the device to choose. If all is working as expected, the name of your 10.3.9 computer should appear in the right window of the chooser.
This section of the same page discusses initiating File Sharing from an older Mac:
http://www.atpm.com/network/files/file_sharing.htm
One problem with the AsantéTalk is that it does not re-acquire devices that drop out. So when you think everything is all set, but nothing appears, the last step will be to reset the AsantéTalk (or cycle its power).