We have written in previous HOTD about the views of modern orthodox rabbis regarding the celebration of different American holidays (see here). All rabbis are very strict in forbidding, for example, the celebration of Halloween in any way, while most would not oppose (and some would even encourage) the celebration of Thanksgiving.
The difference between Thanksgiving and Halloween is that the later (i) has a clear origin in pagan culture, and (ii) some of those customs are still practiced in its celebration today.
What about New Year's day?
According to Christian tradition, January 1st, is the day of the circumcision of Yeshu (eight days counting from December 25), when his name was given to him. Five centuries ago, the rabbi Terumat Hadeshen and the Rama, both living in Christian countries, classified New Year's day as a religious gentile holiday (Darkhe Moshe and Rama, Yoreh Deah 148:12). Terumat Hadeshen refers to January First as "the eighth day of Christmas." He clearly viewed this holiday as 'religious' in nature.
Other Rabbis, however, have a more lenient view, because in their opinion New Year's today has lost entirely its religious overtones and can be rationally explained as a celebration of a new civil calendar's year. Rabbi Moshe Feinstein (Iggerot Moshe, Even Haezer 2:13) writes with regard to New Year's: "On the question of celebrating any event on a holiday of Gentiles, if the holiday is based on religious beliefs [such as Christmas], such celebrations are prohibited .... even without intent, it is prohibited because of marit ayin . . . The first day of the year [January 1] and Thanksgiving is not prohibited according to law, but pious people [ba'ale nefesh] should be stricter [and avoid the celebration]." Following Rabbi Feinstein, Rabbi Michael Broyde (see below) asserts that the status of New Year's day has changed in the last three hundred years. In contemporary America there is little religious content on New Year's Day, and while there might be many problems associated with the way New Year's is celebrated (drinking, etc.) he thinks that few would classify it as areligious holiday, since there is a clear secular reason to celebrate the beginning of the new calendar year.
Most community Rabbis I know would oppose to celebrate, and won't promote any commemoration of the New Year's eve or day. If not for its religious content, for ḥuqot hagoyim (See Rabbi Hofman's article). But at the same time, based on the above mentioned considerations, they won't actively preach against its private celebration by individuals, as they do with regards to Halloween, for example.