Othello and The Crucible are still relevant reading today because they both deal with themes that are important in our society: faith and trust. In Othello, Iago pretends to be faithful to Othello so that he can plot his scheme behind his back. Similarly, in The Crucible, the girls manipulate the faith and trust of their community while innocent neighbors suffer as a result of their accusations. When our collective faith and trust are deceived, our communities suffer because the foundation of our relationships has been tested. Therefore, stories that relate to these themes remain relevant.
Because they deal with themes and topics that still interest and intrigue us today. Humans are creatures of habit, but by studying other human motives and outcomes, we may be able to learn from the mistakes of others and avoid making similar ones in our own lives.
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 as staples for IB and AP EEnglish classes and mandatory reading, they offer readers, viewers, and students alike various themes that deal with everyone's way of life. They deal with life's problems and how to deal successfully with those problems. For when you are in the same situation, and you know how it turned out for the characters, hopefully you won't make the same mistakes.