Under the U.S. Constitution, Congress has the power to enact a fiscal legislation - called appropriations bills - that execute a budget, which is also written by budget committees in both chambers of the legislative branch. However, the President can veto appropriations bills that carry out the budget blueprint, a power that does give her or him significant influence over the process. In addition, the President suggests an annual budget for the federal government to Congress, which also gives him or her the opportunity to shape priorities and the debate over federal fiscal priorities.
As to the question of how much Congress would need to change allocated spending to override a veto, that depends on which party controls Congress - remember, the two chambers could also be controlled by different parties - and which party controls the White House. When one party is in control of both the executive and legislative branches, it may be significantly easier to reach an agreement on a budget.
In the 1996 and 2013 examples, Congress and the White House were controlled by different parties. During the fiscal year 1996 budget cycle there were two U.S. government shutdowns because of the difficulties that Congress and President William J. Clinton had in reaching agreement on all appropriations bills. During the fiscal year 2014 budget cycle, which began in calendar year 2013, there was a 17-day shutdown, again due to the time it took for Congress and President Barack H. Obama to reach agreement. In both cases the President's veto authority likely did drive compromise by Congress.
