From the documentation:
The $.browser property is deprecated in jQuery 1.3, and its functionality may be moved to a team-supported plugin in a future release of jQuery.
So, yes, it is deprecated, but your existing implementations will continue to work. If the functionality is removed, it will likely be easily accessible using a plugin.
As to whether there is an alternative… The answer is “yes, probably”. It is far, far better to do feature detection using $.support rather than browser detection: detect the actual feature you need, not the browser that provides it. Most important features that vary from browser to browser are detected with that.
Update 16 February 2013: In jQuery 1.9, this feature was removed (docs). It is far better not to use it. If you really, really must use its functionality, you can restore it with the jQuery Migrate plugin.