I absolutely refuse to let my family get a dog.
What reason do you have for not wanting your family to have a pet dog?
I don't think they're responsible enough to take care of one.
It makes a lot of sense to have this concern, and you probably know more about them than I do, however, if they really want a dog then most people would agree that it is more kind to let them have one. It would be a good trial run for them to learn responsibility so you should give them an opportunity.
They can't handle the responsibility.
There are easier dog breeds to train than most, and I could help you pick one out if you'd like? Your family would be hurt if you outright disallow them to have a pet.