Unlike in Team Fortress 2, there doesn't appear to be class-specific configuration files in the games content directory. So, it doesn't appear that there is an automatic way to do this. You may, however, be able to do it manually:
Let's say every time you switched to class A, you wanted the O key be bound to
+attack. But for the other classes, you wanted the P key to be bound to
+attack. You can do this with an alias. Go into your Day of Defeat content folder (Steam/steamapps/#your-username#/day of defeat source/dod/cfg) and create a file called autoexec.cfg (if it's already there, thats okay). This will run automatically when the game starts. Open up the file, and type (if theres already something in it, just type at the bottom):
alias "+playClassA" "bind o +attack; unbind p"
alias "-playClassA" "bind p +attack; unbind o"
Now, every time you want to play class A, in the console you must type
+playClassA. When you're done, you type
-playClassA. When you use
+playClassA, it will bind O to
+attack and unbind P. When your use
-playClassA, it will bind P to
+attack and unbind O.
Sorry if this isn't what you were looking for, but it's faster than typing the
unbind commands into the console individually.