He doens't have to put up anything. This is a civil case, not criminal. He can simply claim to be broke.
Except that Trump testified under oath at another one of his trials that he has $400 million in cash and is worth billions. If he now claims to be broke, that's a perjury charge.
You cannot "just claim to be broke" when you go to court. You have to be prepared to prove it definitively. If it's proven you lied, you go to jail.
He can joint ownership them, as well as other things that financial lawyers can ascertain and implement.
He cannot do anything if he cannot own property or do business in the State of New York, and he has a big fine to pay. By "owning property", I would include "own shares in a partnership, corporation or other legal entity which owns property or does business in the State of New York.
Donald Trump will be hard pressed to find any law firm willing to do his bidding to undermine the Court's decisions or orders, given how many of his lawyers are now facing criminal charges, disbarrment, and slander judgements for doing Trump's bidding. MAGA - Making Attorneys Get Attorneys.
Trump is currently interview law firms having had 2 lawyers quit, and Trump's latest announcement that Alina Habba will not be handling his Appeal of the Carroll verdict. The last bunch were definitely third rate hacks, and they were inviting law students to "volunteer" for Trump's defence team.
No competent attorney will work for a client who cannot be controlled and keeps going on TV and committing more crimes. I'm sure Jack Smith has staffers who spend every day reviewing every public utterance by Donald Trump in court, on television and in interviews, gleaning more evidence and confessions of crimes every day.
E. Jean Carroll's lawyer said that Trump's obvious arrogance, entitlement, and disregard for the rule of law, or even common decency, played an enormous role in the size of punitive damages, especially his walking out in the middle of her closing arguments. Showing he was above the law didn't go over well with the jury.
The jury was mostly older white men - truly a jury of Trump's peers. These weren't a juries of femisist women or racial minorities out to get Trump. And Trump continued to defame Ms. Carroll throughout the trial and in the courtroom.
Donald Trump is his own worst enemy. All he had to do what SHUT THE FUCK UP, but then he couldn't play "victim", now could he.