As far as I know, the President has the Constitutional authority to admit anyone to the United States he wants to admit. Nevertheless, 25 Republican governors plus one Democrat means that more than half of the top elected officials of the states have said that they want nothing to do with this program.
Now, even if the governors lack final authority to bar Syrian refugees from their states -- as they well may -- they can suspend cooperation with the Federal government in the resettlement efforts. Of course, the Federal government has lots of money (as much as cares to spend, in the opinion of the current administration).
It's not a constitutional crisis, but when the Federal government sets out to override the will of a majority of the states, it's a crisis of some sort. Given that our enemy has specifically and repeatedly announced its intention to use this refugee flood as a vector for infiltration and recruitment, it's not exactly insane to think that this is a questionable idea. Perhaps permanent resettlement in the United States is not the right option. Perhaps victory in Syria, so they can return home, is the better way.