We are producing a document in print and digital versions. The author would like to use a cover graphic in full color in the digital version and the same graphic, in black and white, in the print version (due to expense of full color printing). Is this a poor design decision? If we need to use black and white in a print product, should we limit ourselves to the same in the digital version? Thank you.
This depends on the printer. With the digital version you aren't limited to color I would do a full color design. Based on how this sounds you're asking for a book to be printed. Depending on the printer the Cover is priced differently then the interior. Price will also depend on how you're going to print, quantity and sell it. If I was going to print something I would want the print and digital products to be mirrored so I would do a full color for print.
If you're worried about color, depending on the quality of the printer they will provide a proof or provide the option for a proof at an additional cost. The only printers I see that do not do this are the ones that typically do wholesale printing and not for the consumer. You should also ask what the price is per run or if they do print-on-demand. Run costs are done usually by the thousand but print-on-demand can be one or two at a time.
Either way your print and digital design should match when it comes to the Cover. If they do not it will just confuse your audience when they go to purchase the document. They will be perplexed trying to figure out what the difference is between the digital and print edition and some may think you're trying to cut corners. If you're selling this you could also develop the digital then after accumulating some revenue produce the print.
That is a common practice. I've done that multiple times for projects with clients. I recently worked on a newsletter and the digital version had full color and the printed version was black and white. It did save a good bit of money for us. It depends on what is being printed whether it requires to be seen in color or it will function fine in black and white.