Reinforcement for Ledger Board on Bearing Wall
The 2x4 ledger is sufficient to support your 2x8 joists, but it needs 4 or 5 16d common nails per stud at 16" on center to be able to carry the load if it is only carrying the floor load and no roof load. Typically the framers would have also nailed the side of the floor joist to the side of the wall stud so the the ledger isn't carrying all of the load, but I'm not going to assume they did that.

If you are concerned about it and want to replace it, first add a temporary support (build a temporary wall) to support the floor joists, next remove the 2x4 ledger, and then replace it with a 2x6 ledger.

Fasten the new 2x6 ledger to the wall with two Simpson SDS25312 1/4" x 3-1/2" heavy-duty screws into each wall stud. Put the top screw 2" down from the top of the 2x6, and put the bottom screw 3/4" up from the bottom of the 2x6. 

I'm suggesting to use a 2x6 ledger to give yourself more room for the new SDS screws to work at their full capacity. If you have the room and want to overkill it, then upsize to a 2x8 ledger and put 3 screws into each wall stud.

If it's hard to find those specific screws, you can use any other 3" to 4" screw if it is rated for 350 or more pounds of shear strength.