To be more specific, the budget actually calls for the complete elimination of federal funding of those two think tanks. Wilson Center has a large amount of non-federal funding (and should be able to survive any cuts), but USIP gets nearly every dollar from the US government.
The Wilson Center appears to be sitting on a large amount of money, with recent financial statements showing it has close to $100 million in assets. Financial statements also show that the think tank received $10.7 million from federal appropriations in 2015, an increase from the $10.0 million it received in 2014.
The Wilson Center's federal appropriation represents about one-third of its total annual budget (the other two-thirds is funded through "trust funds"), according to a document outlining what would happen in the event of a federal funding hiatus.
That document says that about two-thirds of Wilson Center staff is not funded through its federal appropriation. [Christopher Ingraham of The Washington Post says that Trump's budget would strip federal funding for 52 employees of the Wilson Center.]
Here is a previous Think Tank Watch piece on how the Wilson Center is funded, and here is a copy of the Wilson Center's budget justification for fiscal year 2017.
USIP is a completely different story, since by law all of its funding for programs and salaries must come from the federal government. Here is more on the USIP situation from The Washington Post:
The USIP, created by Congress and signed into law by President Ronald Reagan to engage in conflict resolution around the world, is among more than a dozen independent agencies slated for elimination under the budget blueprint unveiled Thursday. Axing the USIP would save taxpayers $35.3 million.
Like other agencies on the chopping block, the USIP has supporters in Congress and the military who can be expected to fight for it, and its death is not a sure thing. But the institute, with its mission of peace building, underscores the budget trade-offs the administration is making as it shifts resources from civilian programs to a military buildup.
A nonpartisan and independent institution, it trains U.S. diplomats and members of the armed forces heading for unstable parts of the world so they can be prepared to help avert conflicts before they mushroom. It also trains local “facilitators” to mediate local disputes.
Even the conservative Heritage Foundation, which has called for many of the cuts that the Trump administration has adopted, has said the USIP deserves to survive, calling it a “do-tank” as opposed to a think tank.
The USIP is far better known in Baghdad and Kabul than it is in Washington.
Despite a record of success, this is not the first time the institute has come under financial threat. In 2011, Rep. Jason Chaffetz (R-Utah) and Rep. Anthony Weiner (D-N.Y.) teamed up to argue it be defunded. They co-wrote an opinion column in the Wall Street Journal, calling it “a case study in how government waste thrives.” Ultimately, Congress gave the USIP $39.5 million of the $46.5 million it requested.
It raises roughly $1 million a year for building maintenance, largely by selling books, charging its 300 employees parking fees and renting out space in its airy Constitution Avenue NW building, designed by architect Moshe Safdie, for private parties.
Here is an article from 2011 about how House Republicans tried to cut USIP's funding.
Here is an article about how the East-West Center could be eliminated under the Trump budget.