“Our efforts will remain focused on the Senate,” Mr. Davis said.

But for Mr. Trump, who has only recently begun to build a modern presidential fund-raising apparatus, the choice of Mr. Pence could yet yield financial dividends. Mr. Pence has his own relationships to donors aligned with the Kochs, along with many other big givers who have until now been reluctant to get behind Mr. Trump or contribute to pro-Trump groups.

David M. McIntosh, a longtime friend of Mr. Pence — and also a former Indiana congressman who is now the president of the Club for Growth — said his group was not planning to get involved in the presidential race. But he predicted that Mr. Pence’s presence on the ticket would prompt some club donors to give or raise money for Mr. Trump.

“There will be some who say, ‘I will give to Mike Pence and help the ticket because of him,’” Mr. McIntosh said. “And you’ll see others who will say, ‘I am still not going to support that financially, but will contribute in other ways.’”

In Indiana, where state law puts no limits on the amount of money individuals can give to candidates for governor, Mr. Pence has collected hundreds of thousands of dollars from wealthy businessmen like Frederick S. Klipsch, the retired founder of an Indiana-based loudspeaker company, and John Childs, the Florida-based leveraged-buyout investor. Both are involved in Freedom Partners, and Mr. Childs is a major donor to the Club for Growth.

David Koch is Mr. Pence’s third-largest all-time individual donor, while Mark Holden, the general counsel of Koch Industries, is sixth on the list.

In an interview, Mr. Klipsch said he had enormous respect for Mr. Pence, particularly the governor’s education initiatives, such as expanding charter schools in Indiana.