TLDR: Dismantling all the components of the current welfare state, taxing the system of production, and benefiting from the eventual commoditization of all physical goods and many services.

From the /r/BasicIncome wiki:

How would you pay for it? First and foremost, the basic income is paid for by direct savings of eliminating the waste, fraud, and abuse of the Welfare State. Charles Murray writes, "After a process that has taken decades, the welfare state has severely degraded the traditions of work, thrift, and neighbourliness which enabled the system to work at the outset. It is now spawning social and economic problems that it is powerless to solve."

By completely ending welfare as we know it, "In the United States, a GI (guaranteed income) for all adults aged twenty-one years and older will cost no more than the projected cost of the current system as of 2011. By 2028, [the guaranteed income] will cost a trillion dollars less per year than the projected costs of the current system." Not all would like to go so far as Charles Murray however, and instead believe it would make more sense to keep certain universal government services, like healthcare and education for example.

Secondly, the complete elimination of the Minimum Wage and all associated payroll overheard for businesses. The reason for a basic income that is a fully guaranteed, realistic, living income (see 'How much would the basic income be?') indexed to the real economy is so that these cost savings can all be fully realized and redeployed toward empowering innovation (Christensen).

Of course, taxes on high-end consumption and financial transactions are currently two of the leading methods proposed to make up any gap between the savings gained in completely dismantling the current means-tested welfare state, and a sustainable basic income. Means-testing is a breeding ground for fraud and abuse in any program, and welfare is not immune. Some argue that waste, fraud, and abuse is so understated and invisible, that the gap between savings in total welfare elimination and basic income could be much smaller than presently calculated.

The general idea of fundraising is via taxation. Just as current welfare systems use tax revenue to fund subsidies, the basic income would as well. A simple setup is just a flat tax on income, and/or a flat sales tax. There are a variety of other taxes that could help to fund basic income, depending on the desired secondary effects of the tax. Many European countries use a value added tax (VAT) to positive effect without materially harming consumption. A carbon tax would help to combat global warming as well as providing a new revenue source for basic income. A tax on High Frequency Traders ("Robin Hood tax") would reduce market "flash crashes" without materially harming market efficiency, and a transaction tax on all electronic transactions (APT tax) would tiny per transaction but massive in aggregate. A wealth tax could be more effective in reducing inequality than a traditional income tax. A land value tax (LVT) - taxing the owners of land for its value, excluding any man-made developments on it - would cause very little economic distortion while raising revenue. Many wealthy people earn more from capital gains than income, so raising the level of capital gains tax is likely to produce a lot of revenue. Inheritance tax helps to fight the unfairness of people born to rich parents having a head start in life. And of course, simply raising income tax is always an option.

Recognizing the existence of our common wealth - the property that no one ever made or we all make together - and charging for its use is another method. This is what's known as the "Alaska Model", because it would be similar to how Alaska funds its dividends for all residents, but it would be extended to all the many other common resources like water, air, the electromagnetic spectrum, Big Data, etc. It has been estimated by Peter Barnes that this method can alone provide everyone in the US $5,000 per year.

One other possibility is to include the funding of basic income with citizen centered monetary policy. In a recession, if interest rates are very low and inflation is not too high, but the economy is not growing, the central bank will essentially print money to help increase demand. This has happened in the current crisis; the US Federal Reserve between 2009 and 2014 added $5 trillion to the money supply in quantitative easing (QE). So in certain circumstances, the central bank could print money and cover some of the cost of the basic income for the government, meaning that the government will be free to either cut taxes or increase spending to stimulate the economy without adding to its deficit.

Basically, there are all sorts of underused ways to raise revenue for basic income. No one tax would be able to completely pay for it, but a combination of the different taxes discussed above, as well as the savings from dismantling the current welfare bureaucracy, make it more affordable than it appears. There are a number of studies which have proposed more detailed costed proposals for basic income: United States, UK, Ireland...

No matter how the revenue is raised however, the origins of its surplus ultimately comes from one place, and that place is the same reason the majority of our efforts no longer goes into food production, but into the 98% of our economic activity that now involves everything else. And that is the machine. The machine pays for it.