There are a few possible solutions, or at least guides.
One might be to send the thing as pdf, if the printer can handle that. Some have problems with separations in pdf. If you do this, you can get some control on the thing. If you use the Press setting, and make the PDf compatible with Acrobat 5 or higher (depending on what the printer uses) and then go to advnaced tab and Transparency Flattener/high resolution things generally work out ok. Acrobat 4 does not support transparency well, so don't use that.
The above sometimes solves the problem of the printer's rip, not being able to handle the ID transparency well. I have several printers where pdf works great, but have problems with ID. I also have the reverse.
When printing from ID there are a few things to check on the Print window. These apply equally when "printing" a postscript file. In the Graphics tab select All under Images/Send data (the drop shadow is a raster - as would be seen in the transparency flattener preview, if raster is selected to display). Another important thing is in the Advanced tab. Make sure that High is selected in Transparency flattener.
If all fails and you have to get the job out, you can try diong the text/drop shadow in Photoshop at a high resolution, make a tiff and stick that in the ID doc. That's the old, tried and true, method.