This is exactly what is done...and then the border is trimmed away after tacking on a piece of dry mount tissue, or the border overmatted up to the image edges. It is known as a 6x9 print (or 6-1/2 by 9-3/4 if you stretch it to the max), and it is how I was taught to print as a beginner.
I can't believe how many people let paper sizes dictate their print dimensions/ratios. Lots of people routinely crop away large parts of their 35mm image, "destroying" their original composition just so it fits a common size of enlarging paper for no good reason. Its nuts! If you are working for someone who requires a print of a certain aspect ratio, so be it. You know that when shooting, so you can compose for it. It makes perfect sense in that case...but if you are shooting "fine art", this makes no sense whatsoever to me.
I print either on 8x10 or 11x14 paper from all formats, and if the negative is rectangular I make the image with a 1" border on both papers, which gives me the same aspect ratio on both types (3:4, which I really like whatever 2F/2F says). So 6x8" or 9x12". It makes the image easy to mount and over-mat without having to dry-mount it (I'm trying to religiously simplify my approach and I really don't need to own a dry-mount press).
Then I use 16x20 ArtCare frames with UV glass for everything and cut my own over-mats to fit. Sometimes, if the image is large enough I can re-cut the included over-mat to size.
If it looks good cropped to fit a piece of 8" x 10" paper then that's how I print it. Otherwise I will print it full frame onto 8" x 10" as Mr 2F/2F suggests.
Usually 20x25 cm paper, with an app. 2 cm white border on all sides. 13x18 to an increasing extent, have begun to like the smaller prints (curiously coinciding with my step up in negative size, apols for straying from the OP).
Paper price plays a role, I would like to print bigger, but have to consider the incremental cost of stepping up in size. Being an amateur, I have no way of passing on the cost.