The correct answer to this open question is the following.
The policy that encouraged the growth of American influence without the use of military power is the Dollar Diplomacy.
The Dollar Diplomacy was created by the United States President William H. Taft under the counseling of his Secretary of State, Philander C. Knox. The policy wanted to protect the political and economic interest of the US in some countries. While addressing Congress on December 3, 1912, Taft said that this foreign policy would" substitute dollars for bullets." That was the way to "protect" or maintain control over countries considered friends of the United States.