Kaldi's code lives at https://github.com/kaldi-asr/kaldi. To checkout (i.e. clone in the git terminology) the most recent changes, you can use this command
git clone https://github.com/kaldi-asr/kaldi or follow the github link and click "Download in zip" on the github page (right hand side of the web page)
To browse the model builds that are available (not many), please click on models.
If you have any suggestion of how to improve the site, please contact me.