Your children will have "sign and sail" cards which indicate that they are minors, either by the color, or a line, or a whole punched in the card, whatever. As long as you make them use their own cards (and don't let them borrow yours or hubby's) they should not be served alcohol while on the ship. We had no problem with this on NCL in the past, but my (then) 15 y/o WAS served a "real" strawberry daquiri on Carnival's Fantasy in the past. She ordered it "virgin", but when it came she had me taste it and sure enough the booze was inside.
Just try to keep up with what your kids are signing for (and where - and who they are with when they are doing so) and you shouldn't have any problems. Of course, teens these days look like adults, so you can't blame the cruise staff too much if the kids have their parents' cards and order alcohol (which DOES happen).
Have a wonderful cruise!