I agree that the local party can make some difference, but I don't think it can explain the shift that has taken place in Nassau.
But, probaly, enough to account for much of a few percentage points that distinguish Staten Island (where the party organization is intact) and Nassau.
I'm sure it makes a difference. But there are major demographic differences between Staten Island and Nassau that I'm sure affect it too.
The demographics of Nassau and Staten Island as of 2000 closely mirror each other, actually: (round to tenths of percentages)
Staten Island: 77.6% white, 9.7% black, 12.1% hispanic
Nassau: 79.3% white, 10.1% black, 10.0% hispanic
