The US Constitution clearly lays out responsiblity for financial matters in the government in Article I, Section 8.
Section 8. The Congress shall have power to lay and collect taxes, duties, imposts and excises, to pay the debts and provide for the common defense and general welfare of the United States; but all duties, imposts and excises shall be uniform throughout the United States;As a practical matter, recent presidents have faced budgets that are so late in being passed that for the first few months of the fiscal year the government operates through continuing resolutions (CR). Under a CR, emergency funding is provided to operate the government because the full budget has yet to be passed. In such circumstances, the president's only practical tool in the budget process, the veto, is useless. No president is going to veto a budget that is already 3 months late getting to his desk. It's an act of legislative blackmail.
Here are the Speakers of the House over the last 30 years and the deficits or surpluses they ran in constant FY2000 dollars, in billions, rounded to the nearest billion. Surpluses are in green, deficits in red.
Tip O'Neill (D) - 1977-86: 136, 141, 89, 147, 142, 215, 333, 283, 313, 318
Jim Wright (D) - 1987-88: 210, 211
Tom Foley (D) - 1989-94: 200, 280, 327, 341, 292, 228
Newt Gingrich (R) - 1995-98: 180, 115, 23, 72
Denny Hastert (R) - 1999-2006: 129, 236, 125, 151, 353, 375, 279, 210
Nancy Pelosi (D) - 2007-2008: 162, 409
The only Speaker of the House to consistently reduce the deficit and whose budgets finally produced surpluses was Newt Gingrich. Everyone else, Republican and Democrat alike, borrowed and spent.