In order to remove the caulk and seal it back up you may have to remove the lid. It will probably crack when you remove it. Be prepared to replace the skylight lid with a new one.
Use the "search button", try a search for "skylight replacement" as there is a lot of info on them in this forum including many pictures related to the replacement. Most people replace the factory type like you have with a more substantial skylight. They are available from a company in Dallas, TX.
Someday when my skylights go bad and need replaced I will order one from Maxim Skylights in Dallas. Their phone number is 1-888-222-4898.
Or, you could just find a replacement lid for the type you have, which looks like a standard issue one, from an Airstream dealer.
You'll have to weigh the costs, etc.
Just take a look on line at the Maxim ones available and you'll know what to do.