How much damage did the Cohen cross-examination do to the State's case against Trump?

That all depends on whether or not you believe there is a fair and intelligent jury, a fair but confused jury, or a jury that is in the bag for the prosecution.

So here is my take:

If we have a fair and intelligent jury, then the prosecution case is likely already dead. No fair-minded intelligent person watching this case would find enough proof to convict anyone of the crimes suggested, which is 34 counts of falsifying business records. There is no evidence that Trump even knew how the financial records were being recorded in the books and the prosecution has failed to explain to the jury how the entries were even wrong. The invoices and receipts were hashed out by Cohen and Allen Weisselberg (Trump Corp CFO) according to their own witness. Even if Trump knew how Cohen and Weisselberg decided to invoice this, they have provided no evidence that Trump directed it, much less any evidence that he directed it with any criminal intent. If you cannot prove Trump is responsible for either or those, then how does signing a check become a crime? This jury is likely seeing the red herring for what it is and the fact that Cohen is a serial liar who probably lied in this trial is just all the more reason to acquit.

If we have a somewhat honest, partially fair, but ultimately confused jury who may or may not be predisposed to believe the prosecution, then they are probably coming to the same epiphany that many of our liberal media members are experiencing. If they have been convinced that the case really revolves around whether or not Cohen is telling the truth, then obviously the cross becomes a problem. Cohen clearly lied, and almost (but not quite) admitted to it by suggesting that he "still believes that he probably" talked to Trump during a 90 second call. Problem for Cohen is that there were text messages before and after the call that make it obvious that the call was made to Trump's bodyguard to discuss prank phone calls, not Stormy Daniels. How does he expect anyone to believe that he first spoke to the bodyguard and tried to get him to sic the feds on a 14-year-old kid, and then moved on to discuss the details of the Stormy Daniels NDA with Donald Trump... all in 90 seconds? More to the point, why did he not provide that story to the jury up front, rather than only suggesting it when confronted. He lied. Everyone knows it. This jury probably starts to question a guilty verdict.

Obviously, if we have a jury who simply wants to convict Donald Trump and doesn't care what the evidence is, then they will simply ignore what happened and move forward with their guilty vote. They will continue to buy the story that Trump is guilty, well just because. None of this will matter.

But here is the reality. It is unlikely that the entire jury will consist of any of the three groups. If Trump is lucky there will be at least two or three of the first group and maybe three or four in the second group. If he gets that, then he gets a hung jury. I seriously doubt that there will not be several that fall into the latter category and will reject the notion of an acquittal on the basic notion that this is their chance to send the bad orange man to jail. So, an acquittal is unlikely. But after the cross examination and dismantling of the prosecution's star witness, a hung jury now seems more likely than not.