The amount of platinum needed to mint a coin worth $1 trillion would sink the Titanic.
It's no surprise that Republicans don't know the first thing about money. The linked Yglesias article puts it nicely:
But saying that the government would need a lot of platinum is like saying a $100 bill needs to have 100 times as much cotton in it as a $1 bill.
Let's skip that part. Let's accept the absolutely ridiculous notion that a trillion dollar platinum coin needs a trillion dollars worth of platinum to mint. So, even if we accept that ludicrous notion, it turns out that the NRCC is still wrnog.
Some maths - the price of platinum is somewhere around $1,600 per troy oz but it fluctuates. One trillion dollars at this price is 625 million troy ounces. A troy ounce is 0.0685 pounds so this works out to around 21,000 tons.
The Titanic had a cargo capacity of 46,000 gross register tons.
Edit: It has been pointed out that GRT is a bad measure for how much weight it takes to sink a boat. That's true - it almost definitely requires more weight than this. GRT is a measure of permanently enclosed volume - or the amount of "buoyancy" the boat has. Since one GRT is 100 cubic feet of enclosed air or 2.8 cubic metres - the actual buoyancy is 2.8 tonnes (the mass of water that this air would displace). Additional, register tons have something to do with subtracting "non-revenue earning space" etc. Complicated rules anyways. Basic point - the Titanic had bulkheads and other enclosed volumes equal in volume to well over a hundred thousand tons of water. Another consideration though, the boat itself has mass. The light load of the Titanic (no fuel, passengers or provisions) was a bit under 40K tons leaving at least 60,000 tons of buoyancy. The loaded weight was ~52K tons (light load plus 14K "deadweight" or allowable* cargo load). Even when fully loaded, if one additional passenger in the form of Moneybags McPreciousMetals were to board the ship with his one trillion dollar 21,000 ton platinum coin, the Titanic still floats. And while she wouldn't be safe to take out into the open seas with this load, history has shown that this was never possible in the first place.
*allowable means actual. If she were loaded less than this, they would fill with ballast to get to this load since the ship was designed to be operated at this draught.