The House began voting yesterday on President Reagan's stripped-down budget as Democratic leaders, struggling against a heavy pro-Reagan tide, charged that the president's plan amounted to what Budget Committee Chairman James R. Jones (D-Okla.) called an "executive tool for domination of the Congress."

While Reagan continued his soft-sell lobbying of conservative Democrats at the White House, Democratic leaders undertook a last-ditch campaign at the Capitol aimed at exploiting longstanding congressional fears of executive usurpation of legislative authority.

"We have served in this body a considerable number of years and we can say frankly that we have never seen the House appear closer to voting to destroy its effectiveness than it seems to be now," said 19 Democratic committee chairmen in an unusual joint letter designed to prop up apparently sagging support for a less drastic Democratic alternative to Reagan's budget.

Like Jones, the 19 chairmen were responding to $36.6 billion in program cuts that would be required under Reagan's budget, as revised with White House blessing by Reps. Phil Gramm (D-Tex.) and Delbert Latta (R-Ohio). They complained that mandated cuts would usurp normal legislative authority and added, "We believe it is no exaggeration to say that the Latta substitute would destroy the integrity of the legislative process as well as the budget process."

But the plight of the Democrats, whose control of the House is sorely strained by apparently heavy bipartisan support for Reagan's budget-cutting drive, was best expressed by Speaker Thomas P. O'Neill Jr. (D-Mass.) in remarks to a morning Democratic caucus.

"Am I happy with the [Democratic] alternative program?" asked O'Neill.

"It cuts you off at the knees so you will use a prosthesis, rather than cutting you off at the hip where you have to walk along with a crutch the rest of your life," he said.

"Are we running into the woods because we are frightened, because a man, the president of the United States, is as popular as he is?"

In the only action of the day, the House voted by voice, after a testy partisan debate, to amend the Democratic budget alternative, drafted by the Budget Committee, to raise defense spending authority to $6.7 billion to bring it into conformity with Reagan's proposals for the Pentagon.

Although Republicans castigated the move as a "last-ditch effort to save this flawed program [the Democratic budget]," they let it pass, apparently confident it would not prove to be the sweetener that the Democrats had hoped it would be.

Before recessing for the day, the House began debate on a substitute budget offered by the 18-member Congressional Black Caucus, which would save much of the social spending that Reagan and the Democrats propose to cut while balancing the budget by closing many so-called tax "loopholes." It also includes a tax cut but targets it more toward low- to moderate-income taxpayers than Reagan's tax cut does.

"It does the best job of eliminating deficit spending, reducing the level of federal spending, providing tax relief to the American people and spurring jobs and economic recovery for American business and American workers," argued Caucus Chairman Walter E. Fauntroy (D-D.C.).

"If we can vote to reduce school lunches," asked Rep. William Clay (D-Mo.), speaking of the caucus proposal to reduce the present business lunch tax deduction, "Why can't we vote to reduce business lunches by 50 per cent?"

A vote on the Reagan budget, technically offered as a substitute for the Budget Committee draft, is expected today or tomorrow.