There are three major "alliances" of airlines: Star, Skyteam, and One World. Each has a website as well as a dedicated forum here at FT. Within an alliance you can usually get miles on your home program no matter which airline you fly. Get to know these families and research how you can travel DEN-BOM on each.

It makes sense to sign up for one---and no more than one---FF program within each alliance. Usually it's best to do so at a US carrier. For example, accumulate miles on United even if you fly Lufthansa.

After that you can choose your flights based on fares or schedule and whichever carrier you happen to be on get miles on your "home" program within that family.

Whenever you can without making other compromises, it is a good idea to accumulate miles in one program.

As others point out, both Star and Skyteam should offer good DEN-BOM options. One World seems weak to me, but you should not take my word for it.