Why are The Crucible and Othello still relevant reading today?
In order to answer this question, it's probably more helpful to look at the themes of the piece, rather than the plot events.
In "The Crucible," Arthur Miller takes on some pretty hefty themes, especially when they were applied to the Red Scare in the 1905s and 1960s. Miller asks his audience to consider when should a person go against his/her moral convictions? Is it appropriate to do so, even in the face of death? Miller also explores the idea of mob mentality--which is even more prevalent today, with the various forms of media we can access and the amount of communication we partake in. E-mail forwards warning about dangers of fake cops or product malfunctions circulate like wildfire through the internet--a modern example of this mob mentality. How many people stop and make sure the e-mail is true before forwarding it? This goes all the way up the scale to the panic about the swine flu.
"Othello", similarily, takes on the themes of doubt and trust. Othellos, being one of the few black people in his society (and in Shakespeare's plays), feels a certain form of alienation and mistrust of those around him. Iago, jealous of Othello, play upon these negative feelings, and trust Othello against his friends, his wife and himself. Everyone experiences these same feelings of self-doubt and mistrust at some point, and most people can relate to being betrayed by someone at some point.
So, the relevance in these plays is found more in the themes and ideas which they explore.
They are both relevant because we are human and we are intrigued by the human condition. In both of these stories, humans are motivated by greed, envy, jealousy, anger, fear, love, etc. How is that any different than humans in today's world? The answer of course, is it's not. This is why the classics are the classics. They deal with themes, emotions, and elements of human nature that will always be relevant. If we don't study them, we are doomed to make the same mistakes in similar situations.