Can a lien be put on a jointly own home in New York State

If a judgment is awarded against you in the lawsuit, a lien can be asserted against your interest in the property.  If by "jointly owned" you mean owned as joint tenants or tenants by the entirety, the lien would only attach to your surviviorship interest in the property.   The judgment creditor would not be able to collect on it until and unless the other owners died, although it would present an obstacle if you wanted to sell the proiperty at any time. If, however, you own the property as tenants in common, the judgment creditor could foreclose on your interest to satisfy the judgment as soon as the lien is recorded.