Yes, drill holes in the studs. Use a 2" hole saw to get a 1.5" DWV line over to the sink. That means you need to install metal protection plates on both sides to protect the pipes, too. If you have a drill with a side handle, that will help. I use my hammer-drill with the hammering turned off.
When planning the holes, plan for a 1/4"-per-foot slope so the drain works properly. Be sure to use a true level, like a bubble level, as a reference. Don't simply measure from the floor because floors aren't always level (especially in basements.)
If the hole doesn't come out right the first try, don't be afraid to use a chisel or reciprocating saw to adjust it. You don't need perfection here; you just need the pipe to go through the studs at approximately the right slope. If it ends up having a bunch of play in it, add a pipe hanger to hold it where you want.