While I don't know for sure why Bay has such narrow entrances I think that it was a combination of:
Lack of space - trying to fit LRT entrances around preexisting buildings and roads. This is also likely why the entrances are not directly on Jasper Ave where visibility and common use would be better
Lack of funds - The pedway entrances all face away from Jasper Ave and the stairs go directly down to the connection tunnels instead of coming down directly from Jasper ave and doing a boot leg turn at the bottom that would probably have been sky lighted. With them facing away, the entrances get forgotten about and checks of the entrances require turning up side streets.
Assumptions about usage - Bay station, while quite lovely at station level, has always seemed to be almost an afterthought station. I suspect it was primarily built to serve the Bay department store and only secondarily to accept passengers from surrounding businesses. I severely doubt that anyone thought of it as a high traffic station.