Agreed. The shared vans (Super Shuttle, etc.) are not pleasant. They won't leave the airport until they're full, which means you could spend half an hour circling the airport and loading more people in until they're crammed to the gills. And the drivers are (I've found) usually more nausea-inducing than cabbies. And since they charge per person (though there's a discount for multiple people in the same party), it's probably going to be about the same cost for a cab or car service.
Public transit to Long Beach is available and cheap, and while it's what I would do as I'm far too cheap to pay for a taxi, it'll take you at least an hour and fifteen just to get to downtown Long Beach and then another half an hour on a city bus to your hotel. That said, if money is a consideration and you want to do that, you'd follow these steps:
1) Take LAX shuttle bus G to the Green Line/Aviation Station
2) Take the Green Line to Rosa Parks
3) Transfer downstairs to the Blue Line (lift available)
4) Get off at Long Beach Transit Mall (last stop, though the train makes a loop, so it won't look like the last stop)
5) Walk one block south and board bus C
6) Ride to the Queen Mary and then back towards downtown Long Beach; get off bus at your hotel (will be on right shortly after departing Queen Mary)
Again, though, while I would do that, I probably wouldn't recommend it for your tired family of 4 with luggage.