With the exception of Magneto and Cyclops, all the mutants have their faces uncovered. So, it shouldn't be hard to recognize these people especially when they were in the News Channels as Wanted Fugitives all the time. How did they hide their identities then? Also, being in the Wanted list has never stopped them from going out or any other civilian activity?
TL;DR For the most part, the X-Men have always worn masks. Recently their identities have been made public and they don't have a need to hide.
In the comics, the X-Men almost always wear masks.
Original X-Men (1963):
Giant-Size X-Men (1975)
Of these lineups, only Storm, Banshee and Rogue consistently don't wear masks. Nightcrawler used an image inducer to make himself appear human.
Starting in the late '80s, Roma granted the X-Men the ability to hide from all electronic perception, including photography and cameras. This was during their time in the Australian Outback after they were believed to have been killed in a battle with the Adversary in Dallas, Texas.
Outback X-Men (1988)
All of the X-Men pictured above have been granted the ability to hide from electronic perception. This electronic deception protection, at some point, went away (this probably warrants another question). The '90s X-Men still wore masks.
Mutant Genesis X-Men (1991)
Of the ones not wearing masks, Rogue was most likely still protected by Roma's magic and Psylocke had gone through a body makeover. I'd imagine Jean Grey's identity was protected by the same logic that keeps Clark Kent's identity secret (even if her code name was her actual name). Same for Gambit.
Starting in 2001, Grant Morrison outed Professor X as a mutant, and the school as a school for mutants in general and the X-Men specifically. At this time, the X-Men stopped wearing masks, most likely due to the public knowing who they are.
New X-Men (2001)
In the comics today, the X-Men publicly live on the island nation of Utopia, and they are public figures. Most of them are known to the public (like Cyclops, Emma Frost, Dazzler and Beast) and even if not, having "normal" lives and civilian identities are not part of their plans. They're currently a full time mutant preservation force. If you want to find out if a heroes identity is publicly known, the Marvel Wiki has that information on every hero.