Hi Pam,
You can do what you propose, except you need to be sure that all the fixtures are vented properly or the PTRAPS could end up getting siphoned off and sewer gasses could enter the house, OK? Here, if the setup you plan is like the first picture on the left in my picture, you CANNOT do this as the fixtures are unvented (I'm betting that the laundry sink doesn't have a vent...it should).
Ideally, the laundry sink is vented, the washing machine drain would be piped and vented and the two basement vents would connect together and connect into a 2" FUTURE VENT provided at many basements nowadays (look for FV written on a pipe in the basement ceiling). Otherwise, by code this combined vent is supposed to go vertical and connect into the upstairs vent a minimum of 6 inches above the flood level rim of the kitchen sink or the vent can be run up and into the attic and connect into a vent up there (see middle picture).
If that doesn't float your boat then you could also install mechanical vents (AAVs) at the basement sink and at the new laundry drain...AAVs must remain accessible for future change outs and you should know that the mechanical vent (called an AAV) is illegal in most states. You never know though...could be legal in your state...
See last drawing.
You may just be better off adding an AAV to the basement sink and draining the washing machine into that!!
Anyway, that's how this breaks down. Glad to answer questions, OK?