As of July 1, 2018, the US population stood at 327,000,000.  Of these, only around 22.4% were under the age of 18. [Ibid.] This leaves approximately 240,000,000 Americans over the age of 18.
For purposes of simplicity I will not be factoring in the several million Americans imprisoned. Any UBI scheme would have to answer the question "Would they be receiving UBI too?". The most straightforward solution to this would be to deny it for all months in which the subject spent any time in jail or prison. Again, for the sake of ease I will assume that all American citizens in the appropriate age bracket will be receiving UBI every month of the year.
$1,000 x 12 =$12,000 per annum (year) for each eligible person. 240,000,000 x $12,000 = $2,880,000,000,000. Or, that is to say, 2.88 Trillion dollars. That would be the rough price tag of UBI every single year, not counting such things as administrative costs.
In comparison, the International Monetary Fund has estimated that total GDP in 2019 would stand at 21.4 Trillion.  What this means is that in order to pay for Universal Basic Income, the United States would have to spend equivalent to over 10% of its total GDP every year.
At this point, my opponent would likely step in and say "Aha but you're wrong! You see, UBI can replace many existing welfare programs!" And in fact this is true...but only to an extent.
In 2018, welfare spending by the federal government stood at $851.1 billion dollars.  Note that this doesn't count the services of the Department of Veterans Affairs, because, well, the general tendency is to think of the VA as just compensation to brave soldiers who put theirs lives on the line rather than as a government handout. In 2019 the White House requested $83 billion for the VA.  We can combine this to get a rough estimate of $900-950 billion spent on welfare annually.
Even if UBI completely replaced all of these programs (doubtful, especially in the case of the VA for reasons already stated), that would still leave an extra 1.9 Trillion dollars. The government would have to collect the money to pay for this somehow. If taxes weren't raised, then this would all become deficit spending.
In 2019 government outlays (expenditures) are projected to stand at 4.4 Trillion, while receipts (amount collected in taxes) will stand at 3.4 Trillion. [Ibid, page 121]
Whenever outlays are larger than receipts, the government borrows to pay the rest. This is called the deficit. The deficit describes how much debt the government is going to take on that year, which is added to the outstanding National Debt. In 2019, the deficit is projected to be almost 1 Trillion dollars; if you were to add the costs of an implemented UBI, you'd have a minimum deficit of 2.9 Trillion.
(Please note: Elderly people have a strong tendency to view Social Security as something that they've earned/invested rather than as welfare. If you deduct a retiree's UBI allowance from their Social Security payout, they'll perceive that the government is either robbing them of the money they earned/invested into the system or discriminating against them by denying them the same UBI younger people get. There are tens of millions of people over 65 in America, and they consistently turn out in large numbers come election day. If it's even hinted at that either party would try this, I absolutely guarantee you they'd lose in a landslide. That is to say, the elderly under this scheme would receive UBI on top of Social Security, whether you like it or not.)
At this point you have two options:
First, you can accept a 2.9 Trillion dollar deficit. This is clearly unsustainable, given that the US national debt is already larger than GDP, and that the deficit in this scenario would significantly outpace economic growth (see projected GDP growth rates in source 2).
How large is the national debt already? 23 Trillion dollars, as of the time of posting. 
Second, you raise taxes to such a degree as to keep the deficit manageably low. The fact that we have such a large deficit already suggests this isn't going to happen, because the political will is lacking.
In addition, tax rates affect investment, wages, and so on. There is a phenomenon called "Capital Flight" which can have devastating effects on any economy, and which is much more likely to happen when tax rates are oppressively high. All in all, talking about this would entail an entirely separate debate that I'm simply not ready to have, because I'm not an economist. Suffice to say, however, deciding that you're going to raise another 1.9 Trillion dollars in taxes every year would have drastic and far-reaching consequences.
With that having been said, I'm going to move on to my second point.
2. Price Inflation
Imagine the following scenario.
You own a supermarket. All of the sudden, your typical customer has $12,000 more dollars every year in disposable income. In addition, taxes on your business went up to pay for this. Way up.
So what do you do? You jack prices on everything. It's clear that they can afford it now, so they probably wouldn't care that much. Even with these price increases they still enjoy greater financial security than they did before UBI.
And this would affect everything. Absolutely. Everything. There is no consumer market whose prices would not skyrocket in the aftermath of this. In the end, you would merely be subsidizing the likes of Wal-Mart.
What would the end result of this be? While people would still gain from UBI, the free money would not go as far as you might initially think. Relying on a cost-benefit analysis (the price of a 1.9 trillion dollar outlay increase vs. the benefit to the consumer who has more money), the balance would not be in favor of benefits.
3. Perverse Incentives
While in the future work may become obsolete altogether, at present the economy still relies on human labor.
However, there are millions upon millions of people who hold jobs out of necessity, because the alternative to working is living on the streets.
But what if all of these people got a paycheck of $12,000 a year? Something that they would receive regardless of whether they worked or not. There would be stigma attached to them receiving UBI because everybody else would. In this case, you could have groups of, say, 2 or 3 slackers who get together, pool their UBI money, pay rent on a cheap place, and live off of that without having to work another day of their lives.
Without these people, the economy would possibly contract, making the above-mentioned deficit problem even worse.
My opponent cited Alaska as a model for the US to emulate. The problem with this is, the economy of Alaska does not mirror that of the US. As his own source admitted, Alaska is an extraction based economy. The oil can be extracted and sold by anyone, and this will happen whatever the tax rate is. The US proper, on the other hand, is a wealth creation economy. High taxes stifle this process.