Mr. Steyer has not yet named the candidates or the State Senate districts he plans to target in Washington, but his strategists are eyeing about half a dozen key seats that could tip the majority of the State Senate in favor of Mr. Inslee’s agenda. Right now the Senate has 24 Republicans, 2 Democrats who caucus with the Republicans and 23 other Democrats. Mr. Steyer has not said what he will spend in the districts, but his previous pattern indicates it will be hundreds of thousands of dollars for each candidate — a huge amount for a Washington State race.

In 2013, Mr. Steyer’s group spent $275,000 in Washington to help ensure the election of four pro-environment candidates to the local council in Whatcom County, where candidates normally spend less than $25,000 for a single race. As a result, prospects are now dim for council approval of a proposed $600 million port in Whatcom near the border with British Columbia, which, if constructed, would facilitate the shipment of 48 million tons of coal annually to Asia from Montana and Wyoming.

The port would be a lifeline for the states’ coal producers, who anticipate the shutdown of American coal-fired power plants because of emissions limits proposed by the Obama administration and hope to send their coal via rail to the proposed Washington port. But Mr. Inslee and Mr. Steyer see them as a climate disaster in the making.

Mr. Steyer’s money has not always bought results. In 2013 his group also spent $250,000 in a special election to try to oust Jan Angel, a Republican House member who opposes Mr. Inslee’s climate plans, and was running for a state Senate seat last fall. But Ms. Angel raised money throughout the state and won the election.

“Thank God that the people of my district were smart enough not to be purchased,” Ms. Angel said. “When you have people with deep pockets like Tom Steyer coming in and trying to trash candidates, spending this kind of money, it’s a sad day for our democracy.”

Others are equally outraged. “It’s ridiculous that money coming from outside the state is trying to influence our votes,” said Rick Tjoelker, an auto mechanic in Lynden, Wash., who bristled at Mr. Inslee’s climate campaign.