Figuring out which version of DirectX you have installed is a simple process. Simply click the Start button, click Run and then type dxdiag. Inside of the main System tab and at the very bottom of the list, you should have a field reading DirectX Version. Beside it, you should see which version of DirectX you are running.
Downgrading DirectX is not something which is recommended by any means. Microsoft does not currently offer a way to remove or downgrade DirectX and therefore users should (NOT) rely on third party software in order to do it. With that said, you can try DirectX Happy Uninstall to downgrade, backup or remove your current version of DirectX.
We highly recommend using third party software to uninstall or modify anything related to your current DirectX installation.
In the odd circumstance that you have installed a game which is asking you for an older version of DirectX than what you have installed, you should think twice about running that software in the first place. Newer versions of DirectX are able to take advantage of most, if not all, technologies used in older versions. If you're running (for example) DirectX 11 and a game requires you install DirectX 9.0c, then there's very little you can do except for using a third party application (see above) to revert back.
In any case, a game which requires you to downgrade a version of DirectX is not worth the money you paid (or didn't pay) for it.
The version of DirectX for Windows XP SP3 is DirectX 9.0c. Unfortunately, at the time of writing this brief, you are not able to upgrade your DirectX version if you're running anything but Windows Vista or Windows 7. In general, Windows Vista comes packed with DirectX 10, while Windows 7 comes with DirectX 11. At this time, you are not able to upgrade from DirectX 9.0c to DirectX 11.