As residents of Westchester County there are many options for assisted living and nursing home facilities in nearby CT. However, there are tax reasons you may want your loved one to reside in New York. In New York all Social Security Benefits are subtracted in arriving a taxable income. CT taxes a large portion of the benefits. New York has a $20,000 pension exclusion, per recipient, CT has none. In addition should the senior be receiving a Teachers Retirement from the State of New York, the pension is completely tax free in NY, where as CT would tax the pension. I see that effective Jan 1, 2019 CT does allow for a partial exclusion. I will have to research if it applies to New York Teachers Retirement Benefits.
For these reasons it may be preferred to be a New York resident.