Acid catalysed hydration of allene
Good question!  As the following figure illustrates, there are 2 ways we can protonate an allene.



(image source)

The top line in the drawing starts by protonating at the central allene carbon.  This generates a very unstable primary carbocation.  Initially, this primary carbocation is not stabilized by the adjacent double bond.  It is not until after this carbocation rotates 90° that its p orbital is lined up with the p orbitals in the double bond and allylic stabilization can occur.  Since there is no immediate allylic stabilization when the carbocation is formed, and since it is a primary carbocation, the pathway leading to it is very high energy.

On the other hand, protonation at the terminal allene carbon (bottom line of figure) produces a relatively stabilized secondary carbocation.  Formation of a secondary carbocation is a much lower energy process.  Therefore, protonation at the terminal allene carbon is lower energy and is what will occur.

Edit: Response to OP's comment


  I always thought that a positive charge appearing on sp3 carbon (which
  then becomes sp2) is more stable than a positive charge appearing on
  sp2 carbon


Generally we think of vinylic carbocations as high energy.  That is true, but what is meant by "vinylic" carbocation is different from what we have here.  Here there is an empty p orbital.  In a vinylic carbocation the empty orbital is not a p orbital, but rather something like $\ce{sp^2}$ or $\ce{sp}$ hybridized.


  is the secondary carbocation more stable because of inductive effect,
  the rotational barrier in the primary carbocation, or both


The secondary carbocation is more stable for the same reasons that we have the  tertiary > secondary > primary carbocation stability order.


Inductive effects, $\ce{sp^3}$ and $\ce{sp^2}$ carbons are electron releasing towards an $\ce{sp}$ carbon due to electronegativity differences
Resonance effects from hyperconjugation involving hydrogens on the carbon next to the carbocation center.