The US was not really in a position to intervene in Mexico initially as it was preoccupied with the Civil War. Certainly they did not support it, and Congress in 1864 passed a resolution to that effect stating, "the Congress of the United States are unwilling, by silence, to leave the nations of the world under the impression that they are indifferent spectators of the deplorable events now transpiring in the Republic of Mexico; and they therefore think fit to declare that it does not accord with the policy of the United States to acknowledge a monarchical government, erected on the ruins of any republican government in America, under the auspices of any European power."

As the war was ramping down and after it, the US began actively supporting the Mexican liberals under Benito Juarez and began pressuring France diplomatically to abandon its imperial ambitions. This was in fact one of the major reasons why Napoleon III gave up on his invasion, as he preferred to keep positive relations with a resurgent US then continue in the seemingly endless conflict halfway across the world as the Mexican liberals put up a bitter fight, and starting in 1866 the liberals began making real military gains. In respect to your question, the US sent a small force of a few thousand men to assist the Republicans around this time as well to show that it was intent on enforcing the Monroe doctrine.

Napoleon III eventually abandoned Maximilian, who made a somewhat pointless and unintelligible last stand against Juarez's forces, but he was ultimately deposed and executed in 1867.

Hope that helped.