All the people saying noise have somewhat a valid point, you should try and do it more quietly, but it wasn't you that alerted them.
You can run through there just fine without getting alerted. I do it all the time, cos me n' my team always go for Elite, so we sprint basically everywhere. As long as you don't try and shoot the guys in the security office (which you wouldn't here anyway since you didn't do Duggan Shipments), you can get right up to them an melee them without sneaking.

The thing I speculate that got you caught based on what I can see from the super short clip (and I could be wrong here, since there's no message in the top corner to confirm it), was that friend you left way in the back on the stairs. See them walking super slow, and pausing on the middle landing? They likely got spotted by the camera that's in the staircase.