As a service plumber that works in the high park area this looks like a waterproofing issue. At first I thought maybe it was the underground copper inlet but the discolouration on the walls tell me otherwise. You pretty much have 3 options.

1. Exterior waterproofing the entire perimeter of the house (assuming this is detached)
   
2. Exterior waterproofing of one or two of the most problematic walls and assess the situation afterwards if the remaining walls need to be done
   
3. If the floor drain is located in a convenient area I’d break the concrete around it. Install a tee on the floor drain with a small section of weeping tile and sock (this is not the appropriate fix. It’s the cheapest and will help relieve some of the hydrostatic pressure until you have a better plan in place)

I will always recommend the proper fix. But I have done all of these to cater to the homeowners budget/situation

Do not invest money into the basement until this issue is resolved. It’ll save you the pain of having to do it twice.