You'll fit right in. The Ivy League schools tend to be far more liberal, both in faculty and student body, than they are conservative, and Dartmouth is no exception. But each of the Ivy schools also has conservative students and organizations, and again Dartmouth is no exception.
Dartmouth gets it "very conservative" reputation because it has a small but particularly vocal and activist conservative student group that about 5 -10 years ago started generating a lot of news stories, some of which went national.
The bottom line is, conservative or liberal, you will be welcome and will find like-minded students at Dartmouth.