What are the backboards that some ballet barres mount to? Are they regular planks of wood or something special?

Glad to see you’re putting the safety of your participants first! I would think that a strong, regular plank of wood would work great.

Just to be sure, have you considered calling a local contractor and getting their expert opinion? Also, are you unable to mount the brackets to any of the studs in walls of your studio?

Best of luck to you finding (and completing) the safest way to do this!

Hello Maryam,
That would depend on the type of barre wall bracket hardware being used:
Here are some helpful links:

http://www.alvas.com/barre_adj_brackets_page.html

http://www.alvas.com/wall_mount_adjustable_ballet_barre_installation_ins…

http://www.alvas.com/barre_fixed_brackets_page.html

http://www.alvas.com/wall_mount_fixed_ballet_barre_installation_instruct…

http://www.matsmatsmats.com/ballet-dance/ballet-bar-wall-mounted.html

I Hope This Helps!
Michael

No, I’m sure I can mount the barres to existing studs, I just want the extra safety that I think this will provide.

Good for you, Maryam Diaab; for being extra safe.
You will want a nice smooth surface and the barres mounted into studs with long bolts. Michael Siler’s idea of contacting a contractor is a good idea; but, I would still go with more than the minimal requirements.
Then of course, just like Michael Saiz says, different situations come into play, also.
Doing things the right way the first time saves more than a stitch in time.
Take care.