What is a passage from Julius Caesar that can be related to a present day issue?
I'd suggest that Brutus's soliloquy in the beginning of Act 2, scene 1 raises issues that most people deal with on a fairly regular basis. In this soliloquy, audiences see Brutus's obvious inner conflict: though he loves Caesar, and has "no personal cause to spurn at him," Brutus is concerned by the possible repurcussions of Caesar as a dictator.
These lines show Brutus's struggle to figure out the right thing to do, and it's obvious that the situation is a difficult one for him. Brutus has good intentions; we are aware of his love for and loyalty to Rome, but also of his love for Caesar.
Ultimately, Brutus decides that crowning Caesar will make him too dangerous and resolves to kill him before he becomes too powerful:
And since the quarrel
Will bear no color for the thing he is,
Fashion it thus: that what he is, augmented,
Would run to these and these extremities.
And therefore think him as a serpent's egg
Which, hatched, would as his kind, grow mischievous,
And kill him in the shell.
Again, these lines represent Brutus's inner conflict, and therefore can be applied to the present-day issues that all humans struggle with. While the situation (friend vs. country) may not be one with which many of us will have to experience, the issue of the internal conflict itself is certainly one with which all of us struggle.
I'm going to go with the most obvious and possibly most well known quote from this play:
Et tu, Brute? (Act 3)
This quote, which essentially means, "You too, Brutus?" has become a modern idiom for any major example of betrayal, political or otherwise. Just do a simple google search of the quote and you will pull up tons of hits where this is the title of blogs and articles in reputable journals or quoted somewhere in the body.
Your question says present day "issue," which makes me ask, does it have to be something famous? Betrayal is an issue that my students are experiencing on a daily basis. Can you relate this to something in your own life? If not, certainly you can find something in the media that also fits.
You may receive several answers to this question, but how about this? In Act III, Scene ii, we have Antony's famous speech to the citizens of Rome - "Friends, Romans, Countrymen....." In this speech, we see a very skillful politician give a speech that is powerful, and full of rhetoric. He uses words, like a true politician, to convince the people that Caesar was unjustly killed and the way in which he delivers his speech, praising Brutus while pointing out his treachery, illustrates the power of a politician's words. You could compare this speech to politicians today -- perhaps how they use their campaign speeches to convince people to vote for them. Our current president, for example, is a masterful speaker and very convincing when he speaks.