They typically don't though. Each years budget, is for the most part, set anew. Some parts may have to be funded from year to year but as a whole its an independent budget.
In regard to the actual "budget" that the President drafts and Congress approves, yes, many of those expenses are from year to year and don't carry over in many instances.
But I'm referring to the "budget" in terms of what we actually spend each year and what we make each year. There are expenditures that do not appear in the official budget that the President drafts, such as expenses for the wars in Iraq and Afghanistan. The defense spending figure we see in the annual budget doesn't include money spent on ongoing wars, at least to my understanding.
Additionally, if Congress were to create a new mandatory program, or alter a current mandatory program, those don't require the passage of annual appropriations bills to alter spending. Mandatory programs continue to spend according to the terms of the authorization bill that was signed into law, and they'll only stop spending or alter their spending of money when Congressional action is taken to alter or remove those programs. Only discretionary programs require an annual appropriation bill in order for spending to be allowed in those programs.
Either way, it's Congress that needs to act on this. Sure, the Presidents are at fault for drafting the budgets and continuing unnecessary spending, but Congress is also failing to act. If a President were to veto legislation that legitimately could help us get out of this hole, then I'd be fine with placing all of the blame on him. But when he's doing what he can to make Congress act on an issue in which they're the only entity that can act, and yet they're not acting, something has to give.
I think it's also worth mentioning the tactic used by Reid and Pelosi of passing continuing resolutions.
To some degree, those continuing resolutions could be needed. When Congress fails to create appropriation bills for existing programs, we have to have some form of legislation which would allow us to spend money on those discretionary programs. Otherwise, existing programs that need funding will fail, as happened in 1995.
I honestly haven't read much on what Reid and Pelosi are doing with continuing resolutions, so I don't know whether they're abusing them or not, although I wouldn't put it past either of them to do so. I just know that continuing resolutions have frequently been used when Congress fails to timely create a budget.