There must be earth religion still around in the Trek Universe. Here's why:
1) In the TNG episode: Sub Rosa, the funeral most obviously has Christian design to it and is obviously just outside an old church, built by the colony.
2) Dr. Phlox mentions observing Tibetan Monks in their monastery, and attending Mass at St. Peter's Basilica in Rome.
3) Christmas is obviously still celebrated as seen in many episodes/scenes in TOS and in the Nexus scene when Picard first arrives in Generations.
4) Kirk mentions, in the TOS, when they defeat the deity-being, that "mankind has no need for gods. We are adequate with the One".
5) In the Kirk-era film Star Trek V, Sybok clearly states that "God does exist. He has many names." and that "Heaven exists. The Vulcans call it Sh-Ka-Ree."
6) A ship's chapel is seen in TOS several times throughout the series.
7) In DS9: The Ship, the Catholic holiday of Carnival is still obviously observed.
8) Amazing Grace is played on the bagpipes at Spock's funeral by Scotty in Star Trek IV.
9) The Biblical book of Genesis is the reference point behind the Genesis Project, the rapid creation of life from nothing (that ultimately failed and was mothballed by the Federation as a result).
10) TOS: The Way to Eden, a group of fanatics believe in the Biblical teachings of the Garden of Eden and believe that it was never on Earth but actually another planet, and that they needed Kirk's ship to find it.
11) In various episodes/scenes in DS9, Voyager, and TNG, Jewish people are mentioned by name, seen in holodeck programs. Worf's adoptive parents' last name and background implies that they are Jewish.
12) The Vulcan salute (double V sign) is actually derived from a Jewish benediction hand sign.
So as you can see, religion, or at least MANY aspects of it survived into the 23/24 centuries. Granted, most religious texts of many of the world's modern religions clearly do not mention beings from other worlds, and most believe that Earth is the only planet in the universe with life on it. If beings from other planets were to land on Earth, I would think religion as a whole would die a very slow death. Probably religious wars would ensue, due to conflict among secularists who would want peace with the aliens, and religious elite who would want nothing to do with the aliens because it would threaten their positions of authority on the Earth.