My husband and I went on Disney cruise 2 years ago. We NEVER had any issues with kids at all. 1 time two teenage girls were running down hallway laughing. No one else was in hallway and they were not doing anything mean, rude or offensive..Just being teenagers.
In fact never heard or saw any kids screaming, crying or throwing a fit. Kids on this cruise were unbelieveable good..At times you almost forget it was a Disney cruise.
Like Cruise Fanatic stated there are many "adult Only" places on ship (and on Disney's own island).