This information comes from a Freeman article by Patricia Doxsey: Ulster County takeover of Safety Net costs would hit some taxpayers hard.

Ulster County is the only one in New York that asks its municipalities to pay their share of Safety Net costs. The amount each municipality pays is proportional with how many benefit recipients live in that community. Money are raised by way of local property taxes.

A move by the state to change the cost-sharing formula from the historic 50-50 state-local split to one that calls for the local government to pick up 71 percent of the costs has prompted county officials to once again consider assuming those costs.
…
If the county bore the entire local cost of the program, it would be applied to the county tax bills of residents in all the municipalities, even those with no Safety Net clients.
…
The Safety Net welfare program would have cost taxpayers in most Ulster County communities more last year had the county assumed the entire local financial burden of the program.

– Jon Dogar-Marinesco