If you can find a DOS version of the "tee" utility (Wikipedia says maybe 4DOS has one?) you might be able to use it to redirect debug's output to a file - e.g. "debug hello.bat | tee out.txt". Then you can use debug as normal, and all text will be saved in out.txt. You can open out.txt in Notepad and copy from that if you need the text.
Alternatively if you can boot a real copy of DOS, you can use the CTTY command to send all output to a serial port, and accept input from the serial port. If you configure DOSBox to redirect that serial port to a TCP connection, then you can use a terminal emulator like PuTTY to access the DOS command line, where you can run debug and copy the output. You'll need to boot real DOS for this, as DOSBox's built in emulator doesn't support CTTY.