Cheap breadboards with steel contacts will rust - check with a magnet! If you hot rinse, shake out as much water as possible then flush with IPA and dry quickly they should be OK. However I doubt its cost effective to do so as most such breadboards deserve no more attention than widlarising them on a 100 lb anvil!
A bigger problem is that although a few brands of breadboard use a hard plastic molded backing plate, mechanically fastened, many breadboards use a self-adhesive vinyl or foam rubber backing sheet to retain the contacts. If your breadboard has a stick-on backing sheet, I doubt it will survive cleaning well as its likely the glue on the backing sheet will let go in places trapping conducive contaminants between the contact strips. You'd need to peel the backing sheet, use goo-gone or similar to remove adhesive residue, then put the breadboard in the cleaner, and fit a new backing sheet made from self-adhesive plastic after the breadboard is clean and fully dry.