That's because it's not in the Constitution, much like the Hastert rule which Boehner claims is causing the impasse. The difference is, the House holding the power of the purse is an old and established tradition, whereas the Hastert rule is just some crap that's been disowned by its namer.
Like a lot of government-in-practice, the House holding the purse strings is just something that's accrued over time. Budgets originate in the House. That's what they do, one of their major functions in modern American politics. It grants them a lever against the Senate and the Executive branch, either of which would otherwise outclass them.
The Senate could come up with their own budget and try to pass it, but that would simply never happen - no one in the House would ever vote for a budget that originated in the Senate, because it would be basically agreeing to let the Senate steal some power from them.