Put together a detailed proposal outlining the form your festival will take and how much funding you require to make it happen, and approach arts funding bodies. Organizations such as the National Endowment for the Arts award funding to cultural projects and events around the country. You can also approach regional funding bodies in your area like the Mid-America Arts Alliance or the New England Foundation for the Arts. Each will have a different procedure for applying for grants but the competition for funding will be intense.
Although it's a concept that goes against the very fiber of many artists' beings, corporate sponsorship of festivals and shows can make artists' work accessible to members of the public that might never come across it in any other way. Depending on the size of your show, consider approaching local businesses that might be interested in being associated with an arts festival. The marketing opportunities available from being connected to a good cultural event are well recognized in the business community.
Philanthropists, the estates of art dynasties, and other wealthy individuals might be a source of funding if they're sympathetic to your cause. Approach individuals or private foundations that have an established link to the area of art that your planned festival is linked to.
If you're confident that your festival is going to turn a profit from gate receipts, commissions earned on work sold, and refreshments and can back this up with a convincing business plan, approach your bank and ask for a business loan to cover the upfront costs of putting the festival on. Persuade the bank of your idea's potential and it will help if you have some sort of arts-related entrepreneurial track record. Failing this, you could approach friends and family for help.