Both as a teacher and as a camp counselor, I have dealt with plenty of separation anxiety in older kids.
In early elementary kids, it is still common to have a transition period as a child enters a new classroom. Even if the child was perfectly happy in the classroom next door the year before, he may spend the first couple of weeks crying in his new classroom. Intellectually, he knows he was happy the year before and will probably be happy again, but in between then and now, he has spent a lovely, long summer in the bosom of his family. For him separation anxiety is wrapped up in feeling uncomfortable with a new routine. Once he has cycled through the weekly schedule a couple of times and feels he knows his teacher, he is fine.
A different kind of separation anxiety, especially when it comes to going to sleep away camp, has less to do with fear of the unknown and more to do with the fear of life falling apart at home. That might sound funny, but at the bottom of a lot of intense homesickness is the sense that "Without me there, my family won't know how to function." In a way, that is quite a lovely notion: A child is so confident that he is an essential part of the family that he feels the family will not be able to function without him. He wonders who will bring in the mail? Who will throw the ball for the dog? Who will keep the baby entertained in the backseat? Ironically, this can come from the parents doing such a good job of making the child feel special, he feels it is wrong for him to leave. Parents can help with this kind of homesickness by having the child be part of making a plan for taking care of anything the child sees as his role. For things like "Who will tell Mommy I love her?", he might brainstorm a solution like leaving Mom a note for every day he is gone.
The hardest kind of separation anxiety comes when there is some kind of tension or trauma in the household. You'd be surprised how many parents who are on the verge of divorce send their children to sleep away camp thinking that a couple of weeks without the children will give them the time to make arrangements for finding a second place to live, etc. Even if the kids have gone to camp before and loved it, this year they bring with them to camp extreme uneasiness about what is going on at home--even if their parents haven't said a word to them--that makes it hard for them to take up life at camp. They are too worried: Are mom and dad fighting? Will they both be there when the kids get home? Will they announce they are getting divorced? Believe me, if there has been tension when you drop your kids off, they will guess the worse--that their parent or grandparent is dying, that the family is going to have to move, that their mom is pregnant. Children who are worried for their parents and about their own security when they return home are the ones whose homesickness does not abate over their stay at camp.
The last kind of separation anxiety is really not the kid feeling he can't live without his parents; it is his fear that his parents have no life without him. I recently answered a Quora Question from a 21 year old who was worried how her parents were going to handle it when she went to study abroad. She knew they were going to be sad, and their sadness was making her sad. My sense was that her parents had built their entire lives around her, and she was feeling the burden of being their raison d'être. When we make our children the sole focus of our lives, we put too much pressure on them. Instead of our being there to care for them, they now feel they have to care for us, to wrap our feelings in cotton wool. That is unfair. It is our job to let our children go--to let them know they'll be missed but also reassured that it is their job to spread their wings and fly.