Yes, despite a movie where they fought each other to the near death, Batman and Superman are actually friends in most continuities. In fact, some see them to be as close as brothers.
Now true, Batman was incredibly suspicious of Superman at first, and does fear the day he goes rogue. However, after working with him for years, a strong trust was formed. Both ways in fact. In one comic story, Batman and Superman traveled the world searching for Kryptonite to destory so that it couldn't be used to hurt Superman. At the end, Superman gave the last piece of green Kryptonite to Batman, because he knew that he would only use it if it were absolutely necessary.
Batman also admires how he affects his city of Metropolis. He inspires the people in a way Batman knows he can't. There were always be tension between the two, mainly because of their different ways of approaching a problem. But, when teamwork is needed, there's a reason they're called the World's Finest.