...are the termites gone? It has been treated, right? If not, this will be only the first in a string of unhappy discoveries. You should have had a termite inspection done that looked at all framing exposed to check for termite damage.
Assuming the all clear is given and the buggers are gone, here is what you would need to do.
If you have an unfinished basement, you have better access to the treads (part you step on), risers (vertical piece between stairs that sounds like your "brace"), and stringers (the part that runs the length of the stairs). A carpenter would do this work, or a handyman. It shouldn't take long to repair if the stairs are unfinished and mostly exposed. This sounds like damage to the riser and should only take a few hours to replace.
I put in a whole new set of deck stairs if a day by myself and that included all new stringers and getting everything level and such. The only hitch might be the stringer since all stairs are tied into it. There is also an epoxy resin that could be used if there isn't too much damage to the stringer (it fills in the damaged part and hardens stronger than wood), but the riser should go.
Let's see if this sparks any more info from you. Can someone easily get behind the staircase? Is this a finished set of stairs? Are all the termites gone?