From the 1880s to the 1980s, New York was swingy. The cities were Democratic because of high proportions of Catholics and Jews. Most of the rest of the state outside the cities was mostly Protestant and Republican. These voting tendencies can be seen well in the 1928 map with state capital Albany, Clinton (Plattsburgh), Oneida (Utica), Monroe (Rochester), Erie (Buffalo), and Niagara Counties. Through the Depression Upstate trended Republican, including the aforementioned counties as well as the New York City boroughs of Queens and Staten Island. In 1964, when Lyndon Johnson (D) won every county, New York shifted Democratic and would vote more Democratic than the country as a whole ever since.
The college counties of Broome (Binghamton) and Tompkins (Ithaca/Cornell) trended Democratic, though Broome would trend back to the Republican column after 2000, and vote Republican in 2016.
Clinton County leaned Republican until after the closure of Plattsburgh Air Force Base in 1995. The other two northernmost counties, Franklin and St. Lawrence, also leaned Democratic from the 1990s to 2012 before becoming swingy again in 2016.
On Long Island, Nassau and Suffolk Counties were Republican from the postwar suburban boom until the 1990s when they trended Democratic. Democratic strength peaked in 2000, and then the counties returned to the swingy range. Suffolk went back to a Republican Partisan Voting Index (PVI) in 2012 and became redder in 2016 while Nassau, which has a slightly higher proportion of non-whites, became a little bluer again.
Westchester was a classic “Yankee Republican” county, voting Republican all the way from 1896 to 1988 except 1912 (likely because of the progressive-conservative split in the Republican Party) and 1964, though the county had been trending Democratic since 1980. The Democratic trends in New York suburbs resulted in Westchester flipping to a Democratic PVI in 1988 and rapidly bluing by the turn of the millennium. After stalling somewhat in the early 2000s, Westchester went back to trending Democratic from 2012 to 2016.
Putnam, north of Westchester, trended Republican after World War 2 like many suburban counties. Many communities’ proximities to wetlands and reservoirs limited development, so Putnam remains in the red column. Dutchess County (Poughkeepsie) also trended Democratic though backslid a little in 2016.
From 1984 on the growing Democratic vote in Syracuse made Onondaga County trend Democratic.
Niagara County voted more Democratic than the country from 1964 until 2004 when it rapidly trended Republican and voted Republican in 2016. Neighboring Orleans County voted Republican in every election since the party was founded except 1964. Even before there was a Republican Party, Orleans voted Whig from 1828 to 1852.
The counties in the Appalachian part are heavily Republican except Chemung (Elmira).
Here are the PVIs in table form.
Here are the maps in slide form.