How was Panama affected by the "big stick" policy?

In the case of Panama, the "big stick" was the arrival of American troops to maintain order.

In the early 19th century, Americans were competing with the British for trade and influence among the emerging new nations of Latin America.  There was an indication that the nations of continental Europe were showing a renewed interest in the area.  President Monroe, in response, issued the Monroe Doctrine that essentially stated that the Americas were no longer open for colonization and an attack on any one of them would amount to an attack on the United States.  We, for our part, would not meddle in the affairs of Europe.

By the time Theodore Roosevelt became president, the United States was no longer an underdeveloped nation, gratefully receiving investments from abroad.  United States Steel, for example, produced more steel than all the mills of Great Britain.  The United States Navy was second only to that of Britain.  As a people we had come to believe that the time was right for us to be seen and heard on the world's stage.

About 1902, Venezuela defaulted on its debts, and was soon blockaded by elements of the British, Italian, and German fleets.  Shockingly, German ships began shelling a Venezuelan port.  What is more, rumors were afoot that Germany intended to build a naval base in the area.  Roosevelt reminded them of our "big stick" (the U.S. Navy) and the Germans backed off.  The Roosevelt Corollary to the Monroe Doctrine stated that the United States had the right to oppose European intervention in Latin American affairs, and we had also the right to intervene in the internal affairs of American republics if they could not maintain order and their national sovereignty.

If, however, the United States wanted to be taken seriously as a major power, it would have to be able to effectively defend the west coast, and this was impossible if the nation's navy had to make the time-consuming, dangerous transit around the South American continent.  We needed a canal.  The most practical route was across the Isthmus of Panama, then a Colombian province.

Secretary of State John Hay negotiated a treaty with the Colombian Charge d'Affaires, Tomas Herren.  The United States would pay Colombia $10,000,000 and an annual rent of $250,000.  The Colombian senate, however, refused to ratify the treaty.  They wanted $20,000,000 and a share of the $40,000,000 the United States would pay the French interest that had already started the canal.  Representatives of Roosevelt got in touch with Philippe Bunau-Varilla, who managed to have a Panamanian rebellion (there had been lots of them in the past) declare independence.  Roosevelt recognized the new republic in three days and our troops arrived to maintain order.