Both "more than me" and "more than I" are correct. The reason is that "than" can be either a conjunction or a preposition. The final verb "earn" is only implied when you choose to use "than" as a conjunction.
Conjunction: "She earns more than I (earn)." Here we are using "than" as a subordinating conjuction.
Prepostion: "She earns more than me." Here we are using "than" as a preposition. (In this case, the phrase cannot be completed with "earn.")
Here is an authoritative source:
Merriam uses "older than I" and "older than me" to exemplify these two correct uses of "than."
[I noticed today (Nov. 23, 2015) that the content of the Merriam page linked above, has changed since I linked to it.]