What influence did the Populist Party have on American politics?
To me, the major influence of the People's Party (the Populists) is that it did a great deal to lead to the rise of the Progressives in the early 1900s. As an often-quoted line goes, populism
shaved its whiskers, washed its shirt, put on a derby, and moved up into the middle class
When it did this, it became Progressivism. The Progressive agenda was really based on the Populists' agenda, only with much less of an emphasis on farmers and rural people.
In the longer term, you could argue that the Populists' ideas have been very enduring. They have (you could argue) led to modern day politicians like Sarah Palin who believe in the idea of "common" or "regular" or "real" Americans fighting a battle against the elites.
Given its very short life at the end of the nineteenth century and the very fixed ties to the silver standard, the populist party had very little lasting influence on American politics after the country came out of the recession that began in the 1890's. It was the first party to actively include women and also tried to appeal across racial and socio-economic lines but did not entirely succeed.
But the name of the party has stuck around and become a label for any party or action politically that is designed to or thought to appeal to a wide section of the population, hence the term "populist."