Since this question was asked for a condo in New York, NY- I am assuming you are referring to common charges (HOA dues is a term that is common outside of NYC). The missed common charges are the responsibility of the unit owner- however that it doesn't always work out that way. From the point of default until repossession, the rest of the owners will have to cover the shortfall. The bank can foreclose, but so can the condo association. While the unit could be sold via a broker, it could also be sold via auction. In either case, the bank holding the 1st mortgage has first priority of repayment (unless there is some government lien, i.e. an I.R.S. lien and then the bank comes gets paid after the government), and then the condo association- if there is still money left after the bank gets paid. If there is still a shortfall- then the other owners in the building need to cover this amount. All other lienholders can get paid after the bank and the condo association gets paid if there is money remaining.
I am not an attorney- so this is just general guidance, but I would recommend consulting with an attorney for specific advice.