If you want a uniform stripe of light, that's one thing, if you want the stripe to fall off, that's another way. Not to mention there's more than one way to get there.
Try to think of it in terms of what you need the light to do. For instance, you want the light to travel across a plane (the background), in a narrow path. That means you need to channel the light rays into a narrow beam. How do you think you can do that? One way is to use a snoot to make the rays travel a straighter, tighter path. The length of the snoot will make them more or less straight, with longer snoots providing the most control.
Of course, light also falls off, as it travels that path, so using a snoot results in a brighter end and a dimmer end, to the light path.
If you want to even out that light path, and make it appear uniform, then you may need to use two lights and snoots, and have one at each end of the path, so that the path appears more uniform.
Then again, you may need to use a projection light through an aperature to create the pattern you need. That will depend on where you can place the light, etc.
The solution is partly in the details of the job at hand, you see. Saying to use approach A vs. approach B, may or may not solve your problem, because I don't know which will work best for your situation at hand.
You can even solve this problem another way entirely, with a double background, where the first actually has the stripe cut out of it, and another background is lit behind it with normal background lighting methods for uniform background light. The light hits the second background and then travels through the stripe in the first background, to be seen by the camera, as a stripe of light.
Most things in photography are problem solving. You have to think through the problem and find the solution that will work in that situation. The situations vary a lot, and over time, you will find an assortment of solutions to add to your repertoire.