Are you printing from Live Type or non rasterized outlined type, and from Illy?
If so do you use a PostScript printer (driver)?
What happens if you Save as PDF and print from Acrobat/Reader?
Without a PostScript printer (driver), the print quality may be low (if printing is possible at all).
Please provide more info. sonds like your printer (driver) is treating is treating vector data as bitmaps, which results in different screening being used and thus reducing perceived quality. More of a problem with your config, or at least in the way how your printer driver and AI communicate, rather than a general problem with AI. To be fair, though, AI's print function is not that great, so Jacob's suggestion of using Acrobat/ Adobe Reader is something to consider...
Thank you Jacob and Mylenium. I will try your ideas.
For my part you are welcome, archibean. Please let us hear your findings.