You are confusing the two distinct concepts: an appropriations bill and a budget. An appropriations bill is what provides the authority for the government to spend money. Article I, section 9 of the US Constitution prohibits the government from spending money without an appropriation:
No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of the Receipts and Expenditures of all public Money shall be published from time to time.
Appropriations bills must be passed by the Congress and signed into law by the President (or the President's veto must be overridden, if applicable) to become valid. If this does not happen before the previous appropriation expires, then Congress may pass a continuing resolution, which is a temporary extension of the previous year's appropriation. (Despite the difference in name, a continuing resolution is a type of appropriations bill, albeit a limited one). If a continuing resolution cannot be passed, then the government will lose its spending authority and must shut down. In practice there are some exceptions. Congress typically makes provisions to prevent services deemed "essential" from closing down. Some programs are funded with multi-year appropriations. Sometimes only part of the appropriation fails to pass, in which case the shutdown applies only to the part of the government that hasn't been funded by legislation. Whatever the case, whether full or partial, these government shutdowns are big news (in the US, at least) when they happen. They also aren't very common in current US politics. The last was in 2013; before that the last one was in 1996.
A Budget Resolution, on the other hand, is completely different. A budget resolution is a concurrent resolution passed by both houses of Congress, but not submitted to the President. As such, it does not have the force of law, does not provide spending authority, and is not binding on anybody, though it does have some procedural effects in Congress. The purpose of the budget is to provide structure for the process of drafting and passing the year's appropriations bills and (hopefully) to expedite that process. Neither house of Congress is required to pass a budget, and in particular, it is not necessary to pass a budget in order to pass the appropriations bills.
When people talk about the Senate not having passed a budget over the past few years, they are talking about a budget resolution. However, in all of those cases where a budget was not passed, the Congress did eventually pass, and the President eventually signed, a complete set of appropriations bills. Since it is the appropriations bills that give the government its authority to spend money, their passage alone is sufficient to allow the government to continue to operate.
So, to summarize, the notion that Congress is required to pass a budget resolution, or that a budget resolution is needed to authorize the government to operate is a canard resulting from the confusion between two very different kinds of legislation, with two very different purposes. In many cases that confusion is sown deliberately by politicians looking to shift the blame for impasses between the Congress and the President, or between the two houses of Congress. Don't be misled; they're not the same thing.
- Table of Appropriations Bills by Year: The link for each year has all of the omnibus bills, regular appropriations, continuing resolutions, and budget resolutions pertaining to that fiscal year.