There's the rule and then there's reality when it comes to paying for any damage your contractor does to your apartment, a neighbor's apartment or your building's common areas.
The rule is your contractor pays for any damage done to your home. Smart contractors repair any damage (such as cracking in ceilings or walls) they generate in your neighbor's apartment. On this point everyone -- including some contractors -- agrees in a StreetEasy.com discussion. (We say "smart" because in one case, the guy "stepped up and made everything right" covering all plumbing and repair charges, and so kept a client and generated good will and possibly new projects.)
As for damages to any common areas, buildings have requirements for insured contractors (underline the "insured") to ensure that possibility is covered.
The hiccup in all this is when a contractor disputes the amount of damages and refuses to pay.
Consensus is that whoever (unit owner or co-op board) hired him contacts his insurance company (the insurer is named in the alteration agreement) to have the issue resolved.
In reality, one shareholder undertaking a renovation voluntarily put down an extra large security deposit (in case damage was done elsewhere in the building). As for co-op-sponsored work, one board said it had to give written assurance to shareholders that it would promptly reimburse them for any damage regardless of whether it collected from the contractor.