By Richard Rubin

Joe Biden wants to combat climate change, repair the country's crumbling infrastructure, improve education and expand government-financed health care.

To finance that ambitious agenda, the Democratic presidential candidate plans to raise taxes by about $4 trillion over a decade through levies on corporations and high-income households.

But winning the presidency won't be enough to get that done. The Democratic Party also will have to win a majority in the Senate while maintaining control of the House of Representatives. President Trump's weak poll numbers and competitive Senate races make that outcome a realistic possibility for the first time in a decade.

In an increasingly polarized political environment, winning that House-Senate-White House trifecta means everything. It's how Democrats passed the 2010 health-care law without a Republican vote. It's how Republicans passed the 2017 tax cuts without a Democratic vote.

On the other hand, a Biden victory where Democrats fall short of a Senate majority would leave Majority Leader Mitch McConnell (R., Ky.) deciding which bills get a vote. Tax policy would be fought on a narrower field, largely over expiring provisions and retirement-savings incentives.

"A Republican-controlled Senate would thwart most of the Biden tax agenda," said Russ Sullivan, a Democratic tax lobbyist and former Senate Finance Committee staff director.

Mr. Biden doesn't need a filibuster-proof 60-seat majority to raise taxes. Holding 50 seats and the tiebreaking vice presidency makes the Biden tax-and-spending agenda viable through fast-track procedures.

In the aggregate, Mr. Biden's proposed $4 trillion tax increase would begin reversing decades of tax cuts. About three-quarters of the burden would fall on the top 1% of households.

"They're unlikely to raise taxes just for the sake of it," said Cathy Koch, a former Senate Democratic aide now at EY LLP. "If taxes are proposed, it will be to offset programs that advance a goal."

Decisions about which programs they want to create first may take more time than usual. Democrats say their approach will depend on the state of the economy and pandemic in January, as they weigh when to shift from economic recovery to the agenda they had long envisioned. The continuing stalemate over another large economic-relief bill just prolongs those unknowns.

"There's much more uncertainty than ever before," said Rep. Mike Thompson (D., Calif.), who chairs the House's tax subcommittee. "We don't know the full impact of Covid."

How much of Biden's agenda gets enacted -- and therefore how much taxes are raised -- depends on the size of Democratic congressional majorities and policy priorities.

"The marginal members who get you to a majority are not Bernie Sanders," said George Callas of Steptoe & Johnson LLP and a former aide to Republican House Speaker Paul Ryan. "They value their relationships with the business community."

In the Senate, a narrow majority would make the most moderate Democrats -- West Virginia's Joe Manchin and Arizona's Kyrsten Sinema -- the pivot points, deciding how far the party could go. A larger majority would shift that fulcrum to others, like Biden ally Chris Coons of Delaware.

On tax increases, look to areas where there's already consensus, said a senior Democratic aide involved in developing legislative plans. The 2017 tax cuts lowered revenue by nearly $2 trillion over a decade, and while Democrats don't want to undo all of them, the first and second trillion dollars in tax increases will be politically smoother than the third and fourth trillions.

Easier-to-adopt policies include raising the corporate tax rate from 21%, reversing the 2017 tax cuts on individual income above $400,000 and increasing estate taxes. The more a tax increase falls on corporations or the very rich, the more Democrats embrace it. Those changes could raise more than $1.5 trillion.

But some of Mr. Biden's tax increases on specific industries could be tougher to get through Congress. Those include imposing a financial-transactions tax and repealing a provision that allows real-estate investors to avoid capital-gains taxes by quickly reinvesting proceeds of a transaction. And some lawmakers may want to wait until the recovery is on surer footing before imposing tax increases that could slow it down.

Democrats' policy ambitions will be tempered by the party's reliance on the upper middle class for votes and donations. That's already evident in broad Democratic support for repealing the $10,000 cap on the state and local deduction, a move that benefits Democratic voters but disproportionately helps the top 1% of households -- the same people that Democrats say should pay more.

Policies with an edge include those already being drafted into legislation on Capitol Hill. House Democrats already rolled out or passed detailed proposals on infrastructure financing and renewable energy. They support expansions of refundable tax credits for low-income families, where Mr. Biden has been less specific.

Sen. Ron Wyden of Oregon, who would likely run the tax-writing Senate Finance Committee if Democrats win, has been developing a proposal the Biden campaign hasn't embraced yet -- annual taxes on unrealized capital gains. And he doesn't necessarily view tax increases as incompatible with economic recovery.

"You start by saying the wealthy would pay their fair share," Mr. Wyden said in an interview. "My view is: You're going to need revenue out of the gate."

Write to Richard Rubin at richard.rubin@wsj.com