Because they are a very functional (if imperfect) democracy with a very healthy economy, a very strong military and generally friendly to western interests in a critical neighbourhood of the world generally filled with undemocratic states who are less stable than they are. Even an apparently staunch ally like Saudi has been exporting some pretty dangerous ideas and lots of money to terrorist organizations that are hostile to the western way of life.
And at an even more perhaps cynical level I support the existence of Israel as a jewish state because. historically, around the world, jews have suffered discrimination and death on a significant scale for coming up on 2000 years. The existence of the state of Israel offers a homeland for a badly persecuted people and, to my mind, reduces the chances of further violence against jews generally across the globe.
Now it is true that the existence and actions of Israel may have galvanized some parts of its neighbours populations against it, and caused some violence, overall I think the existence of Israel had reduced violence world wide, particularly against a historically persecuted group.
So, on balance, I think Israel is a positive for the world.
And to your comment that "even the most insignificant slight is anti-Semitic", welcome to the modern world. Everyone has a thin skin and wants to win the oppression olympics. Israel deserves their shot at the medal as much as anyone.