With specifics to the case, the crime would be a violation of campaign finance laws that could be charged against Trump. However it should be pointed out that some of the details need to be established, which are not.
First, that Trump has not paid off any women in the past that he has slept with and that Daniels was a unique case and specifically to hush her up in the heat of an election. It is not a crime to pay off your mistress so your wife doesn't find out (so long as you initiated the deal. Mistress saying "pay me or I talk" is a crime, but not one Trump committed). If Trump paid off other women when he wasn't in politics, this is not necessarily a crime (even if it was definitely to hush her up for the elections this time).
Secondly, Donald Trump does need to know that this is a crime. While we know his attorney knew at the time, there is no documentation that the attorney told Trump that what he was asking was possibly illegal. Trump can argue that if he had been told it was a crime he would not have committed it and that he payed his lawyer to tell him these things and his lawyer did not. Intent to commit a campaign finance crime is required for a successful prosecution.
Finally, this isn't likely to undo Trump as the Presidents have violated this law all the time without much call for impeachment. The last (and biggest violator by dollar amount) president to have done this was Barak Obama, who misreported in excess of one million dollars in his first term campaign (Trump is accused of something on the order of $100,00 in dollar amount not reported, for comparison) and was found guilty and payed a large fine. It's likely the worst damage Trump will face from this is public opinion damage, not criminal liability.