Constitutionally speaking, the President only has as much influence over the budget as any other congressional bill. Practically speaking, however, the President wields a lot of power over taxpayer funds. Even though it's Congress's job to pass budget bills, it's the President's job to actually spend that money. And that's where things get interesting.
Prior to 1974, there was no formal budget process, and no clear requirement that the President spend money the way Congress had actually allocated it. The President could flat-out refuse to spend money at all (known as Impoundment), and there was little Congress could do to force him. That all changed with the Congressional Budget and Impoundment Control Act, which formalized the budget process and limited the President's impoundment powers (Nixon was in office and was embroiled in the Watergate scandal at the time, so he didn't have the political capital to resist the bill's passage). The law requires the President to submit a budget to Congress every year, so he still has a fair voice in budgetary matters, but his budget is just a proposal.
Even despite this, the President still has a lot of discretionary funds that he can spend however he pleases. There is also a lot of wiggle room to creatively interpret congressional intent with budget spending. For example, the President can choose to "prioritize" resources on a certain agenda, or "clarify" that funds allocated for 'X' includes (or precludes) spending on 'Y'. This is usually accomplished through Executive Orders.
The reality is that the President and Congress share a lot of power over the Federal purse and both have plenty of room to fight each other over spending. This makes it so the President has a lot of sway over what goes into the budget that Congress passes.