I was taught that manners matter and I am teaching my children to be polite, respectful and aware of others and that they should always apologise if there is reason to; but I will never teach them to apologise for traveling. In recent months there have been news stories about parents who apologise in advance of a flight "just in case". I wouldn't.
For starters, the mere suggestion that a child is going to do something that needs apologising for is appalling. The message sent to the child is that they shouldn't be there, that they don't have the right to travel or that by the very nature of being young they are in the wrong.
When you apologise in advance it's a statement that you expect your child to misbehave or act inappropriately. If they cry on the flight, talk or want to play, that isn't wrong, it's a child acting like a child and there is nothing wrong with that. If they spill a drink or kick a seat, say you are sorry. In my experience most other passengers are sympathetic to little kids and to the ones who aren't, I don't let them get me down.
Children aren't perfect-but neither are adults. By all means apologise when it's warrented but until then let children be children and travel.