I plan to use cuDNN on Linux: how to know which cuDNN version I need? Should I always use the most recent one?
E.g. choosing the right CUDA version depends on the Nvidia
You should use whichever is the latest version of cuDNN supported by your application and platform, since that will have the most bug fixes and enhancements. And yes, cuDNN versions depend on specific cuda versions. That is spelled out in the download page. (You will have to be a registered developer to access that page.)
cuDNN v5.1 has different versions for CUDA 7.5 and CUDA 8.0
cuDNN v5 has different versions for CUDA 7.5 and CUDA 8.0
cuDNN v4 and v3 both require CUDA 7.0
cuDNN v2 and v1 both require CUDA 6.5
All cuDNN versions require compute capability >= 3.0 devices
Again, all of the above information is available on the download page for cuDNN. Previous versions can be downloaded from the link at the bottom of the download page.
In addition to CUDA version requirements, you need to ensure that your GPU has compute capability that is high enough:
Wikipedia's CUDA article has a good list of the various compute capability levels and corresponding GPUs (a similar list is available on Nvidia's compute capability webpage).