It depends what your startup does and your objectives.

Take Twitter, for instance. They built their product around their Web 
API, even their Web application is a client for it now. They let the 
community develop clients for them, which allowed them to grow quickly.

I am a strong advocate of applications built on APIs. I think ultimately 
most applications should be built like this. For Web applications, the 
front-end should never talk to the DB directly. In software, concerns 
should be separate and building blocks should communicate through 
well-defined APIs. At the system scale, building blocks are called 
services and this design approach is called Services Oriented 
Architecture.

The best description of good SOA I know is found in this excerpt of the 
now famous Steve Yegge rant:

  So one day Jeff Bezos issued a mandate.  He's doing that all the time,
  of course, and people scramble like ants being pounded with a rubber
  mallet whenever it happens. But on one occasion -- back around 2002 I
  think, plus or minus a year -- he issued a mandate that was so out
  there, so huge and eye-bulgingly ponderous, that it made all of his
  other mandates look like unsolicited peer bonuses.

  His Big Mandate went something along these lines:

  1) All teams will henceforth expose their data and functionality 
     through service interfaces.

  2) Teams must communicate with each other through these interfaces.

  3) There will be no other form of interprocess communication allowed: 
     no direct linking, no direct reads of another team's data store, no
     shared-memory model, no back-doors whatsoever.  The only 
     communication allowed is via service interface calls over the 
     network.

  4) It doesn't matter what technology they use.  HTTP, Corba, Pubsub, 
     custom protocols -- doesn't matter.  Bezos doesn't care.

  5) All service interfaces, without exception, must be designed from 
     the ground up to be externalizable.  That is to say, the team must 
     plan and design to be able to expose the interface to developers in 
     the outside world.  No exceptions.

  6) Anyone who doesn't do this will be fired.

  7) Thank you; have a nice day!

That being said if you just want to make a website and you're looking 
for your model, maybe making an API will be a loss of time and it will 
be quicker for you to just make an all-in-one application that, for 
instance, talks directly to the DB. You can do that if you know for sure 
that there's over 80% chance that you will abandon or completely 
re-write the product anyway.

tl;dr: (Usually) no APIs in prototypes, APIs in finished products.