Leaving Employment Without Good Cause: An employer’s account will not be charged if:
A claimant voluntarily left employment for reasons that did not constitute good cause, provided the employer responds in a timely and adequate manner to Form LO 400, Notice of Potential Charges.
https://labor.ny.gov/formsdocs/ui/IA318.2.pdf This does sound like he voluntarily left your employment (not eligible to collect on your insurance at all) and did not have enough time with his new employer to qualify. Can you prove that he worked for another employer after he left you, and that was the sole purpose for leaving? Do you have any reason to believe he had "good cause" to leave your employment? There are more good causes than you might imagine.
https://www.labor.ny.gov/ui/aso/Section_1600.htm It looks like you may have received a notice that your future premium may increase based on this employee's claim, but that is a lot different than being assessed a charge to settle the claim. The only time I would expect you to be billed is if you were in arrears on your premium payment or you were being charged a penalty of some sort. At any rate ACT AT ONCE as you typically forfeit the ability to contest these issues in a very short time. If you just let it go, it will probably not be fixable.