A spokesman for Koch Industries, Robert A. Tappan, pointed out that Koch companies employ more than 60,000 people and, despite the tough business climate, have 3,500 jobs open. “Koch, like any other business, has to make difficult decisions to ensure the long-term viability of our company,” he said. “The reality is that we have come through a global recession, and growth in the U.S. continues to be sluggish.” (Koch Industries also says that the people in Mr. Begich’s ad are not refinery employees.)

Democrats are hoping to persuade voters that those arguments are hollow. In West Virginia, they lament the loss of 100 jobs in 2010 at a Georgia-Pacific lumber plant in the district of Representative Nick J. Rahall, a senior House Democrat who has been a top target of Americans for Prosperity.

In Arkansas, Democrats criticize the same company, a Koch subsidiary, for eliminating hundreds of mill jobs. They have begun documenting its cuts in two Arkansas towns where Georgia-Pacific, the maker of household products like Quilted Northern toilet paper and Brawny paper towels, laid people off during the economic downturn. Senator Mark Pryor, the Democrat who is up for re-election, has also been pummeled with Americans for Prosperity ads.

Democrats are working hard to draw the Kochs into a more open political battle and help shape the public perception of them as Americans learn more about the money behind the blizzard of midterm advertising. But the battle so far has been asymmetric. Democrats have used their platform as the majority party in the Senate to drive home the Koch attacks, but conservative groups have far outspent Democrats on advertising, helping to weaken the president and his party early on, much as Democrats did to Mr. Romney in 2012.

A recent poll by George Washington University found that 52 percent of registered likely voters had never heard of the brothers. But among the 48 percent who had, more than half had unfavorable views.

Senator Marco Rubio, Republican of Florida, said he found the line of attack unseemly because Democrats appeared to want to punish the brothers for exercising their First Amendment right to political expression. “What I find startling is singling out private American citizens who have decided to engage in the political process, and basically demonizing them by name,” he said. “I think that is something we haven’t seen in quite a while in American politics.”

Alaska, with its robust oil industry, has become an unlikely place for Democrats to test the template they hope to use on the Koch brothers in other states. Mr. Begich has tried to transform the announcement in February that Flint Hills Resources, a Koch Industries company, would stop processing crude oil at its refinery outside Fairbanks, and thus eliminate 81 jobs, into a campaign rallying cry. He has made the brothers the subject of two of his newest ads on radio and television, and condemned them for the commercials that Americans for Prosperity has run.