I want to draw an illustration of a driveway that is not straight - it has slight curvature to it.

I used the Bezier, and dragged the nodes and achieved the line segment I wanted. I intended to copy the line and paste it parallel 80 px away to create the width of the driveway.

When I tried to stroke paint the Bezier line, it created a "ribbon" effect where it appeared to connect the endpoints and filled that area as opposed to creating a fixed-width line (i.e. one px.)

What is the best way to create curved line with a constant width?

Normally you don't have to do anything. Whatever line you draw with Pen/Bezier tool is a fixed width, by default. There's a chance that you won't see it when you draw it, depending on how your preferences are set. In that case, you would need to do exactly what you tried to do, and give it some color or opacity. I can't quite picture what you're describing, but I know that I often accidentally get the Fill tab and Stroke Paint tab mixed up. They are almost identical, and it's easy to do, at least it's easy for me, lol.

Try this -- go to the Fill tab and click the X button. That will remove the fill, if you accidentally applied a fill. Then switch to the Stroke Paint tab, and click the button immediately to the right of the X button. That should give the path a solid stroke with a fixed width.

If that doesn't fix it, then I'm not sure. The way I read your problem, I guess it could sound like you might have used the Calligraphy tool, instead of Pen/Bezier tool. If that's what happened, then you'll need to re-draw the lines. When you remove the fill and set a solid stroke, if the result is a closed path, that probably means that you used the Calligraphy tool. If you hover the mouse over the buttons for a second or 2, some text will appear in a tiny box, that tells you the name of the tool (in Windows, it's called "tool tips")(I'm not sure if they're called the same thing in other operating systems).

If none of this info helps, could you show us a screenshot?

sounds like by whatever means you have inadvertangly set the fill or stroke to something else from the original. check there on the original and on your send copy of the bezier line, adjusting as necessary.