Ben continues to exist as a ghost that only Klaus can see, thanks to his ability to communicate with the dead. The two are one of the great dynamic duos at The Umbrella Academy.
Klaus meets many ghosts at The Umbrella Academy who still carry the wounds that ended their lives and are apparently frozen at the time of their death giving clues as to how they died.
If Ben died young then it stands to reason that his ghost offers some clues as to how he died but why is Ben from The Umbrella Academy getting old when other ghosts don’t?
The answer is found in a classic on the subject of ghosts: Ben grows old because his unfinished business requires it. In season 1 of The Umbrella Academy Klaus sees many ghosts who, like Ben, have unfinished business.
Ben’s unfinished business is different; he was destined to save the world along with his brothers, and died before it could happen. That means he has to get old and grow along with others to be ready at The Umbrella Academy.
Despite having less screen time, Ben is a vital part of The Umbrella Academy team, and actually he is instrumental in stopping (or trying to stop) the apocalypses of 2019 and 1963.