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I find the live shows always entertaining and I've seen many families attend every night.
Some cruise lines, such as Carnival, have two different types of comedy shows with the same comedian. The early shows will be more family-friendly, while the later shows will be more adult. That may be another option.
If you have a few brave souls with you, Karaoke is also fun!
Some of the kids programs also offer family activities.
If the kids are old enough, you could always book ship tours.
Most of the lines have kid's programs too which have their own daily schedules they send out to the families. Those, like the adult dailies, will state all the activities the kid's programs offer during the day and night. They're also geared towards specific age groups, and depending on how old your kids are, you will get the appropriate daily for them.
I started cruising when I was 5 years old, so I basically grew up on those programs!