As others have pointed out, an API that can service both web and mobile is a necessity.
However, beyond that, the answer depends upon the business.
As a general rule, and given the growth and proliferation of "mobile," many recommend that many teams start mobile. That said, offerings differ and are consumed differently pending the platform and form factor.
With mobile (Native/iOS) you get, perhaps, a higher cost of initial development, but a significant barrier to entry (if you are the first mover), in the desktop icon. (Assuming it's consumer focused), unless its games, most apps tend to generate a higher ARPU on iOS--4 to 5x--over Android.
The responsive question is a tough one because display a responsive site within a native iOS environment is typically not the ideal foot forward. That said, as both a backfill and/or a byproduct of starting with a web site, a responsive approach is a no-brainer.
Depends upon the category, product and growth/funding objectives.
The answers are above are thoughtful and accurate. This is just a counterpoint.