Update:
I now have Reaper (32 bit) running under WINE in 64bit Linux Mint 17 with the RME Digi96 card working perfectly.
First install the card and then install Linux Mint 17 64 bit.
The card should be recognized. Check: Preferences / Sound
Select: Play Sound Through - Analogue Output RME Digi96/8 Pad
Now Select: Test Sound
Clicking on Test Front Left / Test Front Right should give audio output on the RME sound card.
Now install 32bit Reaper in WINE.
In Reaper select: Options / Preferences / Device / Audio device settings
Now select:
Audio - DirectSound
Input device - In: Default
Sample Format 24bit
Input channels 2
Samplerate 48000 Hz
Buffers 8 x 1024 sample
Audio Thread - Normal.
tick Allow projects to override device sample rate
This should now work OK!