Waterbirth is actually in many respects safer. Some of the complications can be managed easier in the water. Of course, you just need to get a midwife experienced in good management of waterbirths.
Drowning... It is like saying that you are afraid that baby will drown in your amniotic fluid inside your womb
. When baby is born underwater, he/she still conected by the umbilical cord and they are not breathing till they are brought out of the water, the contact with air triggers their breathing reflex.
Then both you and the baby can go and rest in bed, recover and relax. Of course once the baby was brought out of the water, THAN you need to be careful not to let his face go underwater again, but usually you would not stay at the tub at that point, and I am sure you will be paying all your attention to your baby so, that would be really unlikely to happen
Just do your research, find a real good provider ( My friend had a waterbirth in Peru with independent midwife), and do what feels right for you. Also be prepared to have a few options and change your mind during labor, so you will be able to do what will feel right for your body then.