Update (September 2013)
It's been nearly two years since I originally wrote this answer, and quite a lot has changed. AFNetworking is about to go into version 2.0 and has a thriving developer community, whereas ASIHTTPRequest is pretty much dead. Additionally, AFNetworking has dozens of great helper libraries and classes available for handling things like OAuth, Amazon S3, and the like. It's so popular that there's several books available on it.
Given this, there's no reason at all to keep using ASIHTTPRequest. My original answer is below, so if you're still interested do read on - just bear in mind I would now recommend without a doubt not using ASIHTTPRequest and going with something more modern (namely, AFNetworking, but other options are available).
Original Answer (December 2011)
How much time have you got? Here's my thoughts, although as a subjective question I don't think there's a right or wrong answer.
I lead a development team that works on commercial apps for sizeable clients, and some of my colleagues are always on the look-out for new libraries that can make things easier. Currently I guess the vast majority of our networking uses ASIHTTP. We took a look at AFNetworking about six months ago, and decided at the time we'd stick with what we know. ASI has been around for a while, and there is far more documentation and helpful advice available online for it. It is a solid codebase, and been around long enough to have gained a lot of traction and respect.
That said (and hopefully people who disagree with everything I've said above will read on!), I think if you're starting afresh now is the time to move on to something else. Even the developer of ASI thinks so. Given all of the changes that have come with iOS 5, I think a modern actively maintained framework like AFNetworking (and there are other options, like RestKit) is a good choice.
Now, you're right to say some people have trouble using AFNetworking in their application. But equally, a quick google reveals many people who had problems with ASI (valid or not)! I think whatever framework you go with, someone somewhere will have had trouble with it, either through something genuine wrong with the framework or their own limitations.
So with all this in mind I think I'd dive in to AFNetworking, and see how you get on! One of my new year resolutions will be to migrate over to a new networking library, and I think it will probably be AFNetworking.
From Author's Comment:
A little update: I've actually really enjoyed using AFNetworking, and in many ways it's considerably better (probably not the right word: 'refined', maybe) than ASI. That said, you shouldn't go into it expecting a 'like for like' replacement, and there are certain things you may want to consider other options for (Amazon S3, which ASI supported out of the box for example). But I'm generally really pleased, and find it a clear, well documented library. So if you're looking to move from ASI to something else, you really can't go wrong with AFNetworking...
