The main threats to a cruise liner are fires and grounding (capsizing). Over the last 10 years there have been several cruise ship fires and probably everyone has heard of capsizing. But such accidents are very rare, given the millions of cruises each year. There is a much higher risk on the way to the airport in a motor vehicle than on a cruise ship.
Strict oversight is provided by the Coast Guard, and regular inspections of all ships, monitor compliance with all emergency control requirements. There is also an international convention for the protection of human life at sea, whose rules govern everything from fire protection to navigation and safety at sea. A few years ago the International Cruise Lines Association passed a bill of rights for passengers of the cruise industry, which is responsible for the comfort, safety and service on cruise ships.
Of course, you have to remember that a cruise ship is like a big hotel complex. You need to keep a close eye on the children, to explain how to behave if they get lost on the floors. Keep all valuables in the safe. Also, always be briefed at the drill, which is held on the day of boarding the liner.