Underground downspout pipe may be leaking
It's not clear what you are telling us.  If the PVC is French Drain type, built with holes, then it's intended to allow water to flow in/out. If it is simply broken, then of course you should replace the broken length so it functions as designed. 

To determine whether it is the source of the leak, do what plumbers do: dump water with some dye mixed in down the pipe and see if the colored water shows up in the basement.  If not, it's more likely what you have is a high water table and a microcrack in the basement floor (which may be patchable with hydraulic cement).

And finally, I dunno where you live but the towns I've been in generally ban any on-property drain from feeding a storm drain. YOu might want to check out the local codes.