DOSBox is using a graphical output, so I don't know if that's possible, sorry. 🙁
But if you're running a rather simple text-mode application, you may want to give MS-DOS player a try.
It is described here: https://www.zophar.net/pc/ms-dos-player.html

Edit: here's the English translation of the author's homepage.

DOSBox SVN-lfn has mouse copy/paste support. I've tested it by pasting something from the clipboard to the DOS prompt by right-clicking, as with the Windows Command Prompt. It works only when the mouse is not locked, though, and keyboard copy/paste isn't available.

Oh, hey, at the bottom of that page is a link to something called vDosPlus, which looks like it's for you... a DOS emulator designed for text-based applications (rather than games). Hey, this is pretty nice. It supports keyboard copy/paste (Ctrl+V works at the DOS prompt). It also looks really nice with its scalable TrueType font replacing the blocky DOS font.

Thanks for the suggestions, folks, I tried both vDOS and MsDOS Player, but neither will run my program. (I wrote it myself in PowerBasic many years ago.) MsPlayer bombs out with a message "error: int 2fh (multiplex interrupt)" and vDOS refuses to run it, saying "No 8087," which, IIRC, is a numeric co-processor. Apparently the vDOS emulation doesn't extend to emulating the presence of that chip. So it looks like DOSbox is the only emulator that will actually run it. I guess I'll just have to live with the limitation of not being able to copy text from the DOSbox program window.

If you only care about the output of the program, you can just redirect the output to a text file by running the program like this:

prog.exe > output.txt
prog.exe >> output.txt

Using the ">" operator will erase the content of the output file, ">>" will add it at the end of the output file without deleting anything.

This should work in any DOS-based command prompt, including DOSBox. The downside is that you don't see any text while the program is running.

Thanks K1n9, I' know about output redirection, but it's just not practical in this case. The output loses all meaning unless it is laid out in a specific visual pattern on the screen. My program is a calendar comparison program. It shows corresponding dates in two different calendars in two adjacent month-grids. It allows navigation by days, weeks, months, years, etc. and this is only useful when output is to the screen, otherwise the user is running it 'blind.' Also, with each navigational 'move,' the grids are entirely 'redrawn' (textually) - It's useless sending that kind of frequently repeated output to a file, especially since it would lose its on-screen layout, and would just be a meaningless linear bunch of text and numbers. Occasionally, I want to copy just parts of the screen output because the program also displays certain allied information like Julian Day Number, the year-type, etc and It is just those bits of information that I occasionally want to copy from the screen for easy insertion into a calculator or to make a note of it in a text file. Redirecting the whole program output to a file wouldn't be useful for that purpose, but thanks for the suggestion anyway.

Have you tried the SVN-lfn build of DOSBox that I linked earlier? I don't know how well the copy/paste support works in applications, but it's worth investigating.

Thank you Osprey,

I did not notice them when I first looked at the page you linked to, but on second visit, I found these two links on that page:
http://www.individual.utoronto.ca/wengier/dosbox.zip and
http://www.individual.utoronto.ca/wengier/dosboxmg.zip
I tried them both and (thanks for pointing this out), if I change autolock to false in the config file, I can actually use the mouse to select text from the screen and then copy it by pressing the Enter key. It works in both of the versions at those two links. Selecting is far from accurate and you can only do it with a left click and a rough swipe, but I can actually select a section of the screen that includes what I wanted to copy!

Thank you!!!

You're welcome. Thanks for the tip about pressing Enter to copy to the clipboard. That was something that I didn't figure out and why I was only able to confirm that pasting worked. I may be able to use that when I need to copy an error message to the clipboard.

CORRECTION: I wrote in my last post "... and you can only do it with a left click ..." That was a typo. I meant to say, you can only do it with a RIGHT-click. (Just wanted to clear that up in case others wonder why it's not working for them.)