I'm making a popper out of basswood. This is my first ever lure build and I was wondering if it is safe to simply screw in the eye screws for the line and treble hooks? Or do they need some kind of glue or sealer?

I would always reccommend some type of sealer around your hardware.  You can use a drop of epoxy, super glue or in a pinch fingernail polish. Once your topcoat is on your bait will be sealed and no water can soak into the wood. If water can soak into the wood then it will swell and damage the finish  or it will rot the wood around the screw eyes and they will pull out, according to "Murphy", it will always be on your biggest fish.

Run the screws into the bait all the way and then back them out.   Put epoxy on the screws and run them back in the holes and your set.   Now coat the entire bait with super glue and let dry.   This will raise the grain a little so light sanding with 320 grit works well.  two coats of super glue will give you a nice surface to paint on.   For a perfect sealer give the bait a coat of epoxy then paint and clear coat.   That's how I do every bait I build.

listen to this guy ^