Delta uses the common-use lounges in BOM for their BizElite passengers and Crown Room Club Members. There are two of these lounges (run by Welcomgroup and Taj Hotels) and are both located in Terminal II-A. The Welcomgroup lounge is located by gate 7 and the Taj lounge is between gate 3/4. Neither one of these has shower facilities AFAIK, but then again I usually shower at home in BOM so I never needed to use it!
There is also the old AI Maharaja lounge by gate 13 in the old Terminal II-B which is not being used anymore, now that AI has shifted their ops to Terminal II-C. Someone may have picked this up since I was last in BOM (May), but I doubt it. This one had very spartan shower facilities but was a little larger than the other lounges and had couches that were SOOOOO comfy.
The new AI Maharaja lounge in Terminal II-C is quite amazing though. If you can somehow wangle yourself into the big one on the upper level, it is quite the equal of any lounge in the world and is especially remarkable for BOM. The smaller one downstairs in the international transit area of II-C is also supposed to be nice, but I haven't used it yet.