I TOTALLY understand being concerned about being around a bunch of screaming kids; however, as others have said,
DCL is GREAT about separating the masses!!
On the two Disney 4-day cruises I have been on, we traveled with a group of six adult friends/no kids. This coming April, we will do the 7-day cruise to celebrate three 40th birthdays, a 10th anniversary, and a 20th anniversary (10 adults/5 cabins/no kids). We have a GREAT time and rarely are around kids at all. We are character freaks so we do all stand in line with kids to get those fun pictures.
In 2003, by total fluke, I took three cruises---two of which were not Disney. On the NON-Disney cruise, the kids drove me NUTS! The kids were NOT occupied and stayed in michief mode. One day the entire elevator kept getting "iced" in butter.........so people in their nice clothes going to dinner were greasy! Kids were intentionally leaving dirty dishes every where........acting like the custodial staff were their yard dogs or something. They ran around totally unsupervised and constantly had elevator races. (I know kids can be naughty on the elevators anywhere, but this was constant.) You would have thought the them of the cruise was "ditch your parents and get laid" because there was much of that going on also. They didn't even look old enough to know about that stuff.
I understand what CM_Mom said about not being worried about show content on a Disney Cruise because on the NON-Disney cruise, my mother kept saying, "Why do all of the dancers have their bottoms totally hanging out?" My sisters and I are not really conservative and we just laughed, but commented that we would not want to explain that to our kids (and the audience was full of kids). Disney doesn't make you have to question whether or not what you will see will be appropriate for the kids.
The ONLY time we noticed "screaming kids" on the Disney ship was one evening when we thought somebody must be killing Mickey Mouse or something......we quickly moved toward the atrium balcony to find out that it was princess visiting time and those were screams of joy. We still laugh about those "stage mothers" who are ripping off their child's Snow White costume, yanking on the Cinderella one, and shoving the child into Cinderella's arms, while getting the Belle costume off of the hanger to yank that one on next.
Next June, my husband and I will be taking my six teenage nieces and nephews on a 4-day Disney cruise (we must be crazy) and one of the rules of engagement will be that they will be given a lot independence, but if I even SUSPECT that they are not being respectful to other passengers or to MY Disney characters, they will meet with Captain Hook and walk the plank!!
You WILL have as wonderful of a cruise as YOU choose to have!!