The education required is dependent on the specific type of budgeting work that is being done, but generally, a background in accounting, economics, statistics or finance would be very helpful.
A person does need to have a least a High school diploma to know how to budget, because high school is really where students are old enough to learn and know the value of money. Learning to budget in high school would help a large portion of the workforce with a high school education on their belt.
A Masters Degree in accounting, economics, statistics or business is preferred. For entry level postions a Bachlor's degree related to any of the above is acceptable.
A good one would be a degree in finance or accouting. Finance teaches you all there is to know about money and finance. In addition to get your degree you may get the chance to intern at a bank, brokerage house on accounting firm.