Mount Washington, the tallest mountain in New England, only climbs to 6,288 feet, not quite high enough to experience any sort of altitude sickness (in most cases). That said, there are still a handful of fairly high ski resorts and there are some ski areas that start higher than you might expect.
New England is full of ski areas, from single, surface lift fed hills to larger resort experiences. Some of the oldest ski areas in the United States are located in New England, in addition to some of the most well known (to be clear, New England is a set geographic and cultural region that includes Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, and Connecticut).