Both “Based out of” and “Based in” are correct but can have slightly different meanings. “Based out of” means that you have a headquarters somewhere but you also work in different places. “Based in” means the majority of your business/work takes place in that place.
I am based out of Austin but I often travel to other states.
I am based in Japan at the moment.
I recommend using “Based in” because it has a clearer meaning. “Based out of” can cause more confusion.
“Based in” can be used when you want to say that you live and work in a certain place. The main function of “to be based” with a place is to mean that this place is central to your operations.
I am based in California.
The company is based in Silicon Valley.
The above examples are clear when talking about a person’s life. They live and work in California. They may go on work trips but the majority of their work takes place in California.
The second example most likely means that the headquarters of the company are in Silicon Valley.
Based out of
“Based out of” is more common in American English.
In military English, “based out of” means that a mission may be taking place somewhere far from their base.
The Fifth regiment is based out of Denver. (Though their operations may take place in other locations)
There is a lot of confusion about “based out of” where “based out of” means that someone is based outside a location(as in their base is somewhere else). This is not my understanding. I would understand “based out of” to mean that their base is in that location but they are often involved in operations elsewhere.
I am based out of Austin.
Means that you are living in Austin but also regularly working outside of Austin.
“Based at” is also possible when you mention a specific place (following the rules of prepositions of place which you can read more about here).
This is not very common as people are usually based in areas.
I am based in London.