If the caulk is dried out then an old knife might not work so good. Maybe try a 1" chisel and hammer. Easy does it. Try using the bevel of the chisel laid toward the wood,,the flat side with the sharp edge might dig in if You don't.
How to use a chisel without gouging is tougher to explain than I thought when I started typing. Try looking at pic 1 and 2 at this link if you aren't familiar with chisels. Removing old dry caulk might be similar to removing the strip of wood in pic 1,,pic 2 shows the bevel.
http://www.rd.com/familyhandyman/content/17903/2/
Sorry I couldn't find better illustrations. Take Your time and steer the chisel handle up and down as You tap it with a hammer and watch close. Final cleanup of dry caulk off wood can be a real @$&(^.
Dean