Science fiction has toyed with the subject of the multiverse, but in only a limited number has it actually approached the subject on a serious level and given it the due respect it deserves. In most cases the stories were written before the authors even knew what the truth of their stories were about, and in others the story was crafted around the actual subject that the theories espouse.
The Wizard of Oz was something of a morality tale that used a tornado to carry a young girl into a fantasy land called Oz. There is much to argue about whether it was a dream land or a parallel universe, because all the people who populated it were related by looks and personality to those she encountered in Oz and the return voyage smacked of a process of waking up more than traveling, but we cite it as a borderline example and let the reader be the judge.
Alice in Wonderland is another old children’s story that speaks of becoming and of not really being. Perhaps the original as written by Lewis Carroll had more the wives tale attached to it, but the newer movie iteration was certainly helped along by the fact that it was a return visit, and definitely accessed by way of some tortured path, potables in liquid and solid form, and a chain smoking caterpillar that eventually became a butterfly. This trip into another world was a learning and growing experience for Alice and to get out it also required the drinking of what must have been a disgusting potion. Again it is difficult to see travel between universe or dimensions require a drug, but then there was that deep hole.
Peter Pan was the land of eternal youth that require “fairy dust” and a lot of flying. In the latest movie a flying boat was the vehicle. The kids had a great time and it was a great story, but aside from the island and the boats flying there was very little to do with a parallel universe.
The Chronicles of Narnia was characteristic of the multiverse because it required movement through a doorway to access and there was no association between the world in the time scale. A minute on earth could be a year in Narnia and so the story shows that while only missing perhaps some months at a time, years and even decades past and things in Narnia changed while only a short time passed in our world.
The Marvel movies that feature Thor use a light bridge to transmit their characters from one world to another. The guard of the bifrost can see the other worlds and watches them, and is the purveyor of travelers. They are moved to and from by this method which definitely is in keeping with the multiverse theory.
In Star Trek there have been a number of stories which have featured trips into alternate realities. One that occurred from a transporter malfunction, one that occurred on a planet surface where a doorway of moving time allowed the user to jump through at the specific time they chose when it was being broadcast in the portal. A similar portal was available to them on a planet where information was archived for universal use.
The One showed an individual killing his own iterations in every parallel universe in order to gain their power. A transporter mechanism was used in this case as well.
Finally, Stargate shows a portal found in ancient Egypt which was used by an ancient alien race to gather slaves and transport them to their planet. These portals were placed throughout the universe and parallel universes so they could travel where they chose at will.