Hi Hi among the jquery ,mootool ,yui and glow have crossbrowser compatiblity
By best you have to define what you mean. It depends on many factors.
I prefer jquery since its user base is too large and the availability of plugins.
JQuery has been gaining good popularity. The compatibilty of JQuery with most of the browsers is higher. Jquery has picked up very well and so, Its better to try out JQuery only.
I use jQuery, becouse it's intuitive and there is a lot of feedback and plugins.
compatibility
Jquery: Firefox 2.0+
Internet Explorer 6+
Safari 3+
Opera 9+
Chrome 1+, There are known problems with: Firefox 1.0.x
Internet Explorer 1.0-5.x
Safari 1.0-2.0.1
Opera 1.0-8.x
Konqueror
Mootool: MooTools is compatible and fully tested with Safari 2+, Internet Explorer 6+, Firefox 2+ (and browsers based on gecko), Opera 9+, and Chrome 4+.
Yui: http://developer.yahoo.com/yui/articles/gbs/
I'd recommend jQuery above anything. Microsoft visual studio 2010 ships with jQuery support which is a good benchmark for this test. Also jQuery has vast audience that develop plugins for jQuery.
At the end of the day, it depends what you want to use the framework for.
I would recommend that you start by learning to write pure javascript, only then will you find out where the relative utility of a framework comes in.
If you want a framework that has the widest browser compatibility, along with best performance on older computers/browsers then the rather annoyingly named 'my library' probably wins out. Simply because it does feature detection rather than try to infer the browser name / version, which is a rather craptastic cop-out that most of the established libraries are guilty of.
I would recommend you give each of them a go and see what suits your need, locking yourself in to one particular framework too early will only hurt you long term; When holding a hammer, every screw looks like a nail etc.
Personally I favor YUI3 - it has strong OOP support as well as flexible CSS3 style DOM querying and a great selection of quality RIA widgets.
I wouldn't let myself be too swayed by arguments ad populem, plenty of people do silly things, find out what is right for you and your problem domain.