OK. Now that everyone has had their chuckles, we'll get to the actual answer. However, the answer is actually dependent upon what you mean by "drive the big boats."
The dude who actually steers the ship is called the helmsman and this duty is done in shifts by several different trades (boatswain is among them). Boatswain's are considered the experts in this area and so when things get tricky (e.g. pilotage and replenishment at sea, when acurate steering is crucial) the "special sea duty" helmsman is almost certainly a boatswain. However, the one who steers the ship (i.e. has his/her hands on the wheel and controls the rudder), is not the one who is actually "driving." The helmsman simply reacts to and carries out the orders of whoever has the "con" (which can be a variety of individuals, all of whom are MARS officers, but is usually the Officer of the Watch or Second Officer of the Watch). In short, it is the MARS officers who are actually "driving" (i.e. they are the ones navigating and deciding where the ship is going). And of course, the engineers are operating the propulsion machinery which makes the ship actually move (although they do this in response to the MARS officers' direction).
If you are referring to small boats such as the rigid hull inflatable boats (RHIBs - which are actually rather big and fast - not little rowboats), then yes, it is the boatswains who drive them. In fact, this is considered the exclusive domain of boatswains (although MARS officers can sometimes drive them as well).