That shot was taken from the retaining wall of the Union Station tracks, which overlooked Aliso Street (now overlooking the 101 freeway and El Monte Busway/San Bernardino Fwy. carpool lanes).
In fact, there's a photo posted on here with a locomotive that broke through it... here, from the LA Times archive:
LA Times
Here's a Google Map view of the retaining wall:
Google Maps
As you can see, an overpass was built through the retaining wall for the Gold Line light rail tracks to cross over the 101 freeway.
Hmmm, I take that back. That old postcard photo was indeed taken from the track area of Union Station, but from a retaining wall that no longer exists. You can see the tiled covered walkway from the postcard here, in this Google Map image, in the lower left corner. You can't get that same view anymore because of the MWD headquarters building.
Google Maps
Here's an aerial image of Union Station from 1940. That retaining wall is near the upper right-hand corner, where the curved ramp for the roof parking is. The old postcard photo was taken from there. The MWD headquarters building now occupies that site.
USC Archive
I remember that ramp. The MWD building was built in the late 1990s.
Undated photo
LAPL