How do the producers of Gossip Girl reveal that they think wealth promotes/allows for social mobility in early 21st century New York? Which episodes demonstrate this idea that wealth allows for you to mover up the social ladder? or vice versa..

First of its mainly the writer that come up with stuff like this not the producers to clarify.
Second, its not saying that wealth enables the climbing of social ladder. It saying that wealth is part of it but mais importantly its a birthright. like take for example, when nate father went to jail, he was still part of the inner círculo because he was born into that kind of family. A family who is extremely privileged and part of that power that they achieve is due to their wealth. It ultimately due to the name they are born into.