This article is from the Amiga CD32 FAQ, by Stuart Tomlinson with numerous contributions by others.
A: Connect a serial link (buy a ready-made link or make your own - see the diagram for the Aux port) between the CD32's Aux port and the other computer. Then run either Sernet or Twin Express on the other computer and the CD32. You'll have to find a bootable CD-ROM with one of these programs on it and you may need to run NoReset unless Sernet and Twin Express are on the same CD as the files you want to transfer.
A: If you have an interface attached that gives you a parallel port, you can connect a parallel link between the CD32 and the other computer, remembering to use a special Parnet cable and not a normal parallel cable (otherwise you could easily fry everything). Run Parnet on the other computer and the CD32. You'll have to find a bootable CD-ROM with Parnet on it and you may need to run NoReset unless Parnet is on the same CD as the files you want to transfer.
ParNFS is a replacement for Parnet, with bug fixes for Workbench 2 or above machines (that includes the CD32). It fixes the Dir, List, and Directory Opus bugs.
A: You can use The Communicator. This is a link from the CD32's Aux port to the Amiga and a CD-ROM with a series of utilities on it...
Communicator... Display text and graphics on the CD32. Play Audio and Video CDs. Transfer files to and from the CD32.
Twin Express... Serial file transfer program, Amiga and PC versions.
Workbench... Modified to allow the joypad to be used as a mouse.
Photolite... Display Kodak Photo CDs.
Scala driver... Allows Scala, running on another Amiga, to control the CD32.
There are two versions available. The Communicator Lite doesn't have a keyboard through port or MIDI in/out/thru.
A: There is a new version of The Communicator, with the original title of The Communicator 2. Improvements to the previous version are...
Virtual CD... Appears to make more than one CD-ROM accessable to the host Amiga, also stops resetting.
File Transfers... More reliable - Zmodem. Faster - apparently it can reach up to 210000 baud. More compatible - works better with ISO-9660 CDs.
AGA Viewer... View pictures and anims in any mode, including AGA. Will also work for non-AGA machines. Supports datatypes.
Workbench... A newer version (doesn't say which).
Mediapoint driver... Alows Mediapoint, running on another Amiga, to control the CD32, including Audio and Video CDs.
Hardware... Added A2000/A3000 keyboard port. Serial lead is DB25 to RJ10 (telephone) and is easier to remove.
The Communicator Lite 2 doesn't have a keyboard through port, MIDI in/ out/thru, or status LEDs.