I prefer no seams in my borders, so i like to use the length of fabric (LOF) which is parallel to the selvage edge, rather than the width of fabric (WOF) with seams.
The trick to cutting LOF is to have your fabric perfectly aligned on grain before you cut.
Start with a length of fabric which is a few inches longer than the length you need for borders, taking into account extra for mitred corners (if applicable) and the fact that the ends of this length will need to be squared up.
Iron your fabric open to one layer. The process is easier, i think, if you spray starch or mist out the fold so that it is open and removed from the fabric.
Then lay out your length of fabric, on a table, preferably.
Take one selvage edge and bring it to the other selvage edge, and make sure the fabric lies perfectly smooth. No little waves or wrinkles. The cut ends of the fabric may or may not be lined up. What you need lined up is are the selvages with the 2 layers of fabric completely smooth. If you wish, you may press the fold back into the fabric at this point.
Then fold the length of your fabric, with the selvages lined up still, in half, and in half again, until it fits under your ruler.
I like to use the June Tailor Shape Cut Ruler, with a large Rotary blade (60 mm rather than 45mm), but you can use any ruler you wish.
Be sure to put your ruler perpendicular to the folded edge and parallel to the selvage edge when you cut, or you will end up with something that looks like ric-rac.
Once you do this a couple of times, it is a piece of cake and you will be able to do it always.
To save on fabric waste, you would need to see if your border fabric is used elsewhere in your pattern and if it is, you would need to figure out how much of the fabric which is not used in the borders would be available for your quilt top. Often you can get away with buying what you need for the borders and then you can get the remainder of the pieces for your quilt from the "leftovers" from the borders. Just remember to cut borders *first*.