Should parents continue to finance their children after the children are 18?

I think it's important that children of that age learn self-support and money management, so no, I don't think you should support them after that age. I suppose it's a little different if they're a University Student, but even then, they can arrange their own student grant and keep books with their budget and spending in. Money management is a very important lesson to learn in life, and so is the commitment to support oneself and one's family. Encourage them to seek work, and put money into the household, whilst they learn about the trials of life.