We can tell that Horatio believes in ghosts—or, at least, that he has come to believe in them—by inference from what he says and does after the ghost appears in act 1, scene 1 of the play. At first, Marcellus states that Horatio "will not let belief take hold of him" and is unwilling to believe the reports he had heard from others. As such, Marcellus asks Horatio to accompany them, "that he may approve our eyes."
When the ghost appears, however, Horatio seems to change his mind. He agrees that it looks "most like" the king and states that it fills him with "fear and wonder." When the ghost begins to leave without responding to Horatio's questions, Horatio prevails upon it, "I charge thee, speak!" Clearly, he believes in its agency to do so; he is convinced that the ghost is "more than fantasy."
Horatio admits to Marcellus that he "might not this believe / Without the sensible and true avouch / Of mine own eyes." However, he has evidently been granted sufficient evidence from his own eyes; he is now convinced that the ghost of "the majesty of buried Denmark" is real.