Senator Bernie Sanders, the Vermont independent and 2016 Democratic primary runner-up whose populist agenda has helped push the party to the left, embarked on Tuesday on a second run for president, in a bid that will test whether he retains his anti-establishment appeal or loses ground to newer faces who have adopted many of his ideas.

A professed democratic socialist whose calls for “Medicare for all,” a $15 minimum wage and tuition-free public colleges have become pillars of the party’s left wing, Mr. Sanders joins the race at a time when Republicans are trying to define the Democratic field and its ideas as out of the political mainstream. In Mr. Sanders, who has not joined the Democratic Party, Republicans have an easy target to try to make the face of the opposition.

But Mr. Sanders, 77, starts with stronger support from small-dollar donors and liberal voters than most other candidates. And he is among the best-known Democrats in a crowded field, as well as one of the most outspoken against President Trump, whom he has repeatedly called a “pathological liar” and a “racist.”

“During our 2016 campaign, when we brought forth our progressive agenda, we were told that our ideas were ‘radical’ and ‘extreme,’” Mr. Sanders said on Tuesday in an email to supporters. “Three years have come and gone. And, as result of millions of Americans standing up and fighting back, all of these policies and more are now supported by a majority of Americans.”