Like many things there may be valid arguments on both sides.
If you leave it on 24/7 you may be subject to lightening strikes, power surges and hackers. If on 24/7 I would highly suggest you do not remain in any browser window including your email client - recent hackers hacked into Yahoo itself and everyone who was logged it at that moment was vulnerable and many got hacked. If on 24/7 you may be able to remotely observe the inside of your home. You may have to pay a few extra bucks per month for the extra electricity. There is always the risk of fire when you are not there although this may be an extremely remote possibility.
You will notice that most light bulbs go bad when they are turned on maybe because of the power surge. If this is true it may be also true for computers. Hard drives and monitor life is measured in part by the hours it is spinning or turned on so if its on 24/7 there may be more wear and tear internally and may not last as long.