Because of some people still struggling with BASS x64 and C#, here is a step by step "tutorial".
I use Visual Studio 2005 for this. Because I'm using a german version of Visual Studio i translated the labels of buttons etc on the fly to english, so don't assume the phrases to be 100% correct. I did my best

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1. Compile your application using the target "Any CPU"
2. Open the compiled .exe file with Visual Studio via File > Open > File...
-> The resources of the file will be shown
3. Right-Click somewhere in the resource view to open the context menu and select "Add Resource..."
4. In the following dialog click on the "User Defined" button
5. For the resource type enter RT_MANIFEST
-> Now the new resource is opened for editing
6. THIS IS IMPROTANT: Use the property editor to assign an ID of 1 to the RT_MANIFEST resource.
7. Select the following manifest and press Ctrl + C to copy it to your clipboard.

<assembly xmlns="urn:schemas-microsoft-com:asm.v1" manifestVersion="1.0">
<dependency>
<dependentAssembly>
<assemblyIdentity type="win32" name="Microsoft.VC80.CRT" version="8.0.50727.1434" processorArchitecture="amd64" publicKeyToken="1fc8b3b9a1e18e3b"></assemblyIdentity>
</dependentAssembly>
</dependency>
</assembly>
8. Go to visual studio and paste the manifest into the new resource.
9. Save the executable and your done.
If it still doesn't work you have to check wich version of the Visual C++ Runtime you have installed. To do that go to C:\Windows\winsxs\. There should be a directory named something like "amd64_microsoft.vc80.crt_1fc8b3b9a1e18e3b_8.0.50727.1434_none_88de292b2fb06019" that contains the vc++ runtime. You may have noticed the "8.0.50727.1434". That is the version that has to be specified in the manifests "version" property.
If you don't have such a directory you don't have the vc++ runtime. Google for "Visual C++ Runtime" and you'll find it.
I hope this helps everyone. If not, feel free to PM me