Recently I got a new Macbook Air, which comes pre-loaded with OSX Mavericks 10.9, but had a lot of issues getting Titanium Studio/Appcelerator Studio up and running. This is a list of items to check and helpful hints to fix any issues you may also encounter.
Open up the Mac Store App on your computer and install XCode. When finished, make sure you open XCode and run it - it will first ask you to accept the licence for the software and then ask you if you want to install the additional add-ons. Say yes and download/install.
First and foremost I generally tend to go and install things straight from the source - in this case the Node.JS website. Don't do this. It seems this current version of Node and Studio simply are not compatible. If you've already installed it, there's a handy script and guide to completely uninstalling Node at http://www.maxwhale.com/how-to-uninstall-node-js-on-mac/
Installation Problems when using Titanium/Appcelerator Studio
You may run into an issue where Studio asks for your system password (three times) and three times it will tell you it is incorrect when it isn't. Then Studio will tell you it's installing stuff but it actually fails. The issue here it seems is that it wants to use the root user and not your current user account (.e.g "boydlee" in my case). The simplest way to fix this is by removing Titanium Studio completely and before attempting to re-install it, open up the terminal and update your root password to match your current user account's password.
$ > sudo passwd root
$ > Password (enter your Admin Password)
$ > Changing password for root
$ > New password: *****
$ > Retype new password: *****
Now if you install and launch Studio again when it asks for your password it will actually work. You should be able to install the Android SDK plus all the Titanium updates now.
Permissions on folders
If you'd tried to install Titanium SDK / Alloy from the command line you may have caused some permissions issues and Studio won't be able to access the Titanium folder under Application Support. To fix, just run a CHMOD command on this folder, e.g.
CHMOD -R 777 ~/Library/Application Support/Titanium
Do the same thing to any other folders that complain about permissions. In particular, if you upgrade from Mountain Lion to Mavericks you may find permissions issues with the ~/.titanium folder. You will also need to CHMOD 777 permissions on it.
Unable to Run projects from Titanium Studio
If your "Run" command is disabled, right-click on the project (in Titanium Studio's Project View), select Properties, then Project Natures from the left hand list, and check the "Mobile" project nature.