I need to calculate the design story drift as per ASCE and would like to know whether STAAD.Pro computes elastic drift (δxe) or the design drift (δx) from Eq 12.8-15of ASCE7-10 ?

What you get from STAAD.Pro is the elastic drift and you need to apply the Cd/Ie factor on top of it to get the design drift as outlined in ASCE. Now instead of multiplying the elastic displacement by Cd/Ie, it may be easier to divide the allowable drift factor by Cd/Ie so that the allowable drift computed that way can be directly compared to the elastic displacement coming from analysis.

For example when you specify

PRINT STORY DRIFT f3

where f3 = The allowable drift factor as per the code provision, you can actually divide the f3 by the Cd/Ie factor so that the story drift check can be done based on elastic drift

Another point to note here is that, at any story level, STAAD.Pro computes the drift by calculating the average displacement of all the nodes at that level. If there is a rigid diaphragm defined, then the drift is calculated at the center of mass of the diaphragm.

Please refer to technical reference manual section 5.42 titled Print Specifications and within that specifically refer to the section titled Printing the Story Drift and Stiffness for more information on story drift calculation in STAAD.Pro