Hi, Todd!
More than likely, you
can purchase Disney Dollars on Disney ships at Guest Services. Yet, the funny thing is...about the only place you can "spend" your Disney Dollars on a Disney Ship is at Guest Services. You see, Disney Ships are cashless. Therefore, you would have to apply your Disney Dollars to your shipboard account in order to pay for things such as specialty drinks like smoothies or coffees, snacks at the theater, games at the arcade or items purchased from the stores on board. I guess you could give them away as tips for room service or in other areas -- but that's just not as satisfying for kids who want to "get" something for their dollars.
I recommend giving your children
Disney Gift Cards which they can apply to the family's shipboard account by presenting them to Guest Services. Then, they know they have a certain amount to spend while on board. We've accumulated allowances on Disney Gift Cards for our four children, and it was an interesting experiment to see who had more money to spend while on vacation. Ironically, it was the younger set despite the fact that the teenagers are paid more.
I hope you work out a suitable arrangement for you and yours. Have a wonderful day and a magical vacation. Bon voyage!
Andi