The United States needs to make it known that they are willing to cooperate with our friends in the Middle East. Anything that fosters that sense of cooperation (rather than invasion) is important in forming positive middle eastern relations. I think being involved with a drill of this magnitude would do wonders for our reputation, especially among the Arab world, where we aren't exactly popular.
As the military drill taking place in Saudi Arabia is among its allies and is to instill the message that the kingdom and its brothers are prepared to confront challenges within the region to promote and keep peace and stability. Our participation would lend no benefit to the countries in question, other than of course further concreting our harmful "Big Brother" reputation.
The US doesn't need to posture or put our military readiness to the test. We already spend more than enough time and money on our military budget. If we're going to put so much stock in our weapons, we shouldn't put them to the test against smaller nations outside of combat situations. Why should we give them insight into our plans and functions?
It's about time Saudia Arabia raised its hand and started an effort to solve the conflicts of the Middle East. Although the US is a global leader in every respect, you can't expect the US to solve every single problem in every part of the globe. Saudia Arabia's proximity to the conflicts and understanding of the issues will allow them to be more effective problem solvers.