Um... there is no 64-bit support if you are running inside Vault Explorer or Job Processor. Those applications are built as 32-bit, which means any plug-ins are also 32-bit.
If you are writing a stand-alone application, then you can control how your app is built. The default mode for .NET is to decide at runtime if it's 32-bit or 64-bit. As long as you stay entirely in .NET, your code should work fine in either mode with no extra work on your part. The problems come if you try to interface with non-.NET libraries, such as COM.
The best advice I can give is to never assume anything about what mode you are running in. Just because you are on a 64-bit machine does not mean the program is running as 64-bit, and just because the program is in Program Files (x86) doesn't mean it's 32-bit. In fact, don't even trust the first paragraph of this article.
So what can you trust? The answer is System.IntPtr.Size. You can check this value at runtime to know for sure what mode you are running in. From there you can write the appropriate logic, throw errors, etc.
Autodesk.Connectivity.WebServices.dll is a pure .NET DLL, so it will work fine in either mode.