I've used Commodore 64 emulators before, but it's been a while. I've read about the transfer cables used to connect a Commodore 64 disk drive to a PC, but never purchased one or used one. I researched them again and come up with the following information and ideas.
Since your Dell Dimension 8400 has a parallel printer port that can be set for EPP or ECP mode, you might want to get an X1541-series cable. These are a family of cables that connect to your PC's printer port and to the Commodore disk drive. The selection guide describes the differences between the various cables in the X1541 family, but the conclusion suggests the XM1541 cable is probably what you want.
Comet.vermont mentioned a USB cable that connects to the Commodore disk drive. The original xu1541 is based on the Atmega8-16 DIP. The xum1541 shown in the video is based on the Atmel AT90USB. However, it looks like both are developing products that you'll have to build yourself, or purchase at a higher price than an XM1541 cable.
OpenCBM is a driver and set of command line tools for Windows or Linux that helps you transfer disk images and files to and from your Commodore disk drive over the transfer cable.
The Star Commander is a Commander-style file manager that can transfer to and from the Commodore disk drive via the transfer cable. It's text file instructions suggest using the OpenCBM driver to use it under modern versions of Windows. So it looks like you'll need OpenCBM anyway if you want to use The Star Commander.
Converting your documents
Once you transfer your OmniWriter documents to your PC, how will you convert them into something usable by modern PC software? I have two ideas.
I found a copy of the OmniWriter manual. It includes a description of the document file format. If you're a programmer, or know a programmer familiar with the Commodore 64, you may be able to use this file format description to convert your OmniWriter documents into something usable by modern PC software.
Find a Commodore 64 disk image of the OmniWriter software. Use a Commodore 64 emulator to run the OmniWriter software and load your OmniWriter documents. Setup OmniWriter to print to an ASCII printer connected to the user port, and setup the emulator to save the user port output to a text file. Then when you print in OmniWriter, the emulator will save a text file for you.
I couldn't find a disk image of the OmniWriter software. I found this thread that shows someone uploaded an OmniWriter disk image to a temporary file host earlier this year. If you have the original OmniWriter disk, this thread suggests its possible to make a disk image of it, but it involves removing the copy protection.
The two main Commodore 64 emulators are CCS64 and VICE. It looks like VICE has options to send printer output to a text file, but CCS64 doesn't.