Both of them are relatively soft woods compared to other typically used woods for instrument building, with poplar being slightly harder than basswood.
As tonewood basswood is known to be mid heavy, mainly emphasizing the mid frequency spectrum, whereas poplar is more similar to alder responding pretty evenly over the entire frequency spectrum, and as such known as properly the most balanced tonewood.
Some descriptions add that while poplar is quite similar to alder regarding tonal qualities it does seem to emphasis the high frequency spectrum slightly more.
I voted basswood, because I personally happen to prefer a mid heavy sound.
Janka Hardness Scale:
Basswood 410
Poplar 540
Alder 590
Mahogany 800
Swamp Ash 1320
Hard Maple 1450
Basswood, Alder and Poplar are all pretty soft.
Alder and Poplar are 50 points from each other on the Janka Hardness Scale and are relatively easy to dent but not quite like Basswood which is 130-180 points softer.
Basswood does not hold screws very well and Poplar and Alder do, they seem to be over the minimum hardness to not strip out every screw that goes into them, where Basswood basses seem to have stripped holes on pickguards and strap buttons all the time.
Basswood is lighter than Alder...Poplar is a bit heavier than Alder..for body balance I prefer Poplar and Alder over Basswood, my Basswood bass is a bit neck dive-y while sitting.
Alder is the most abundant hardwood in the pacific northwest and one reason why it was chosen by Leo Fender over other choices.
Cheap, available, easy to cut and shape, and easy to finish.
I have basses that are Alder, Mahogany, Poplar and Basswood and they all sound good ...I'm not sure body wood is really that important overall to sound.
I like Swamp Ash for a number of reasons, 1320 Janka Hardness, heavier but not crazy heavy, great grain patterns with natural/translucent finishes ,and sounds good.