It's by Ken Thorne, though he's imitating the stylings of John Williams. I find the whole sequence sort of bizarrely charming, in a slapstick sort of way, capturing a lot of the moral simplicity of the old Superman movies. (I haven't seen Batman v Superman, so I'm not going to slam it, though of course the fact that a big Superman fan such as myself didn't see it is itself a slam of sorts.)
One notes the way the names fly up from the bottom of the screen, over the action; I think this is supposed to be somewhat reminiscent of what is undoubtedly the greatest film opening credit sequence of all time, that of the original Superman: The Movie (1978). Seriously, watch it on full screen and turn up the volume:
Chills, man, chills. Every single time I watch it, and for every single name that flies in with that funny little noise. You know I didn't see this movie until I was in high school, but it's seared itself into my brain as if I saw it when I was five years old. I'm not sure why there's the prologue about the Daily Planet's Depression-era reportage, though.
The credits for Superman II (1980) are not quite as awe-inspiring, but playing John Williams's Superman march over a synopsis of the previous film gets me revved up a lot, too:
I also have a fondness for the opening credits to Supergirl (1984), with music by Star Trek's best composer, Jerry Goldsmith. Apparently they were hella expensive: those flying names are physical models:
You can see that they were really trying to create something as impressive for Supergirl's first film as they had for Superman's though, I don't think they quite succeeded.
But by the time Superman IV: The Quest for Peace (1987) came out, everything about the Superman films, including the opening titles, was cheap, even if they did get the composer of the original Star Trek theme, Alexander Courage, to do the music this time. Seriously, the credits look like you could have knocked them out in twenty minutes on an Apple IIe:
I feel like I would be embarrassed if I had seen that in the theater. "This is what I'm paying money for!?" (The movie does have its moments, though. Or rather, a moment. I will always love the bit where Superman says in complete seriousness, "It's common knowledge that you hate children and animals, Luthor." Gene Hackman just shrugs.)
Of course, the nostalgia-fest that was Superman Returns (2005) went back to the original source here, with a set of titles that updated the original with modern effects technology:
Plus some really swell Marlon Brando narration. As this was the only one of these movies to come out during my adult life, it's the only one I got to see on the big screen, and I will always love it for giving me my only chance to see the sheer spectacle of classic-era Superman credits in their intended glory.