Seller; you or your real estate agent should have requested a final water reading, which must be paid "prior" to the closing. If you have no agent please check with the closing attorney.
New owner; All outstanding utility bills "stay" with the property. If no final reading was ordered you will be responsible for any outstanding balances due along with interest, penalties and possibly liens. New owner is also responsible for notifying the Water Division of change in ownership and correct mailing address.