How to stick plastic on to fabric
I assume by "bed base" you are referring to the box spring? 

Take some long strips of cardboard and staple it to the bottom of the box spring, over top of the fabric covering. Use some double-sided carpet tape or another type of adhesive to secure the strips to the cardboard.  

This has the added benefit that it's easy to remove and doesn't damage the box spring.



Alternatively, you could do something to the bed frame itself: add some slats (1-by material), or even use some small corner brackets to add a strip of wood around the inside perimeter, and then attach your lights to that, rather than the box spring.



Also, either way, use some type of strain relief on the cord, so if it gets pulled (someone yanks on it, trips on it, or pulls out the bed without unplugging it) it doesn't tear everything down. This may be as simple as using a zip-tie around the cord and a leg, or a screw-in style ziptie to actually screw it into part of the frame.

Without knowing details about the bed frame it's hard to recommend something more specific.