There are Hundreds of Ajax Frameworks available today. Sébastien Just explains How to choose the right one for yourself.

server independent or not ?
server independent frameworks will help you in mashing-up server technologies and lowly ties your software architecture while server dependent frameworks may increase your productivity but makes difficult major changes to your projects.

structural Javascript enhancements ?
Javascript can become a maze if your team does not share a common development methodology . There are so many ways of doing the same thing ( like creating objects of accessing the DOM ) in JS, that you must formalize the way to do it. Such a framework may help via on-demand Javascript, packaging abilities, enhanced OOP...

re-usability of your written components ?
making use of it shall help you in re-using written components in your next projects.

framework current documentation level ?
often low for most projects, you must take care of it

features you need ?
you may look at your projects needs ( or near future needs ) and see if the Framework can handle them. Some are GUI oriented, special-effects oriented, Javascript centrics, communication only oriented. None will completely cover your requirements. What is the complexity degree in making the framework evolve ? can you add your own features in the framework ? will you help committers ? Can you mashup multiple Frameworks ?

How long will it last ?
Most projects are companies own project spin-off. The Framework evolution depends on their ability to maintain and make the project evolve. A good user community is important as well as talking to developers when needed further functionalities.
Taking care of sponsors and the framework current uses in production web sites, is a good start in guessing if the Framework will last years or will be stopped in six months.

What sort of support ?
Is there a community support ? commercial support ? test reactivity while testing it.

How steep is the learning curve for the framework?
Can your team handle such a technology. Many developers are reluctant to Javascript . Can they handle such a change, is the Framework simple to understand ?

Who are my visitors ?
It seems that a very light framework is required for a public site. If your visitors need to upload a large Javascript API, they may not visit further your website, depending on your page's download times. When working on intranets, or professional services, it may be acceptable that the first access to the application may be longer, moreover when the website is used daily. Also think about on-demand Javascript and Javascript compression.