If you look around I think you'll find some open source or free implementation of QB.
For command line apps the FBIDE that comes with FreeBASIC is good. But it's not a drag & drop designer. There's Firefly that works with FB 0.23. I don't know if it's so great for kids as the WinAPI calls are collected into a wrapper library. But you can drag and drop edit boxes buttons etc.. to do the Gui part.
But I think I would take a look around to see if you can find QB open source clones. I know I've seen them around. I don't know if they come with IDE or any details really. There are open source implementations of TurboVision for Pascal.
But for simple command line apps FreeBASIC FBIDE is fine. You can compile and run out of the editor. And FB does have some "compatibility modes" that help it tolerate QB syntax as well as VB etc..
The online forum people are pretty friendly. They don't rip your guts out before helping with your problem.
edit: if you still have the real Dos QB compiler you may be able to run it on top of DosBox free dos emulator.