9pt seems to be the rule-of-thumb but like most rules, they can be broken when it makes sense to do so.

Do you think 6 or 7 would not print well?

Courier is too thin for 6pt.
It’s thin, even at typewriter size.
You may want to increase tracking slightly with AG at 6 pt.

You can go with smaller type size on a business card than other documents, because the type will be read in proportion to the document. However, “optical” size is still an issue.

You can also go with smaller type size on stationery, because it’s the secondary matter on envelopes and letters. But again, optical size is an issue—the design should look as if it’s scaled for the real world, not something that was created by a designer working at 300% on a monitor.

I agree with Nick's comments.

You also need to consider your audience. Our eyesight tends to get worse as we age, so if your client's customers include many folks over 40, you might want to go a bit larger.

Courier is […] thin, even at typewriter size.

Do you mean Courier New, or do you think that even the Mac version is still too light?

People have used 4 1/2 points at times, but that type size is very small and hard to read. Where space is very much at a premium, one might consider 6 points, but even that is very small.

But I don't see 8 points, or even 7 points, as overly small for certain elements of the design that properly should be in small printing. Those sizes are readable if printed normally.

However, since 12 point type on a computer is usually 10 point type with generous leading, as what I've just said applies to actual physical lead type (remember it?) perhaps 9 point type - or 8 point - is the absolute minimum in your situation.

Courier New has stems widths similar to Helvetica 35.
Mac Courier has stem widths between Helvetica 45 and 55, and would probably have enough heft at 7 pt, but you might be concerned about the counters of /m.

Why are you even asking this question?
Why not just do a few layouts with your preferred typefaces at different sizes, and see which works best?

@Nick Shinn:
Why are you even asking this question?

To me, it seems obvious why he is asking this question. He might produce a business card with six-point type on it, and it might look all right to him - but other people might find the small print impossible to read.

So, either he tests his layout on a large number of other people - some who have vision problems or are elderly - or he tries to find out if there is an acknowledged standard in the industry that covers this concern.

"I can read this" is not the same as "Nearly everyone will be able to read this".

It depends on the resolution of the output device and the medium in which it is printed....Something that size at 2400 dpi on smooth resin light sensitive paper is easy to read by someone with good vision. So it depends on what's printing the output and the type of paper being used.

"So it depends on ..." and lots of other things, too. Such as the color of the ink and the paper. Black on white would usually be more legible in a tiny size than tan on beige to cite an extreme example.

pick something from this family and go as small as you want:

http://www.myfonts.com/fonts/256tm/minuscule/