How do I stabilize the edge of a quilt made on the bias?
 

Someone gave me a partially completed top that I would like to add borders to in order to make it big enough for a comfort quilt. Only problem is that it was made with larger HST that are for some reason cut on the bias. In the past I've been able to baste along the edge but this one is puckering as I baste. Could I maybe iron on a small strip of very thin interfacing? or would we see it later?
Suggestions appreciated.

I think the interfacing would work. I'd cut it wide enough to stabilize the seam.

Another option is to measure the border, pin it to the seam, steam press the bias edge to fit the border. Then stitch with the border fabric on top. This may also be a time for hand basting-extremely rarely used.

Just thought of something else. The cross cut of fabric (perpendicular to the selvage) is a little stretchy. So if the borders are cut that way, the stretch can be used to advantage to fit the stretchy bias. It'll go back to it's normal length. This helps to avoid puckers.

What about stay stitching before adding borders. Just run a row of normal length machine stitches about 1/8 inch from the edge. It I'll end up inside the seam allowance but will help keep the bias from stretching.

A lightweight interfacing ironed to the wrong side of the top most likely won't show at all, so I wouldn't worry about making it really narrow. You would have to be very careful not to stretch the top's edge as you iron on the interfacing.

What I would do is measure the top through the middle in order to decide how long the edge should be. Cut a lightweight interfacing to that measurement (cut down the middle of the interfacing to get two edges exactly the same length). Fold the interfacing in half, then fold in half again, and mark each fold with a permanent Sharpie. Do the same with the quilt top (just keep the permanent Sharpie marks inside the seam allowance, or use a thread tack in an obvious color, or chalk or pins). This allows you to distribute any fullness in the top evenly. Pin the fusible side of the interfacing to the quilt top edge, matching registration marks. If there is a lot of fullness in any area, this is where I would hand baste with a single thread to distribute fullness evenly within the quilt top.

I would pin a length of parchment paper to the ironing board and iron the interfacing on with the wrong side of the quilt top facing up. I'd probably have the interfacing extend a little beyond the top's edge. Just peel the parchment paper off while the paper is still warm, then trim the excess interfacing away.

You can test your interfacing on a few scraps but, I think as long as you use a lightweight interfacing, it is not going to show through to the right side.

I for sure would use my walking foot. When stay stitching around the edge.

I'd start with careful pressing with spray starch.

When sewing a bias edge to a straight edge, put the bias on the bottom. The feed dogs on the machine will take up the difference. I always pin, matching the seams that need to match.

I think you would get the best results from measuring the centers for border lengths, pinning them carefully to the length and width of the quilt. Pinning will help control stretch. Sew with the bias edges on the bottom next to the feed dogs. If you need to hand baste, do so. Some of the fullness in the bias HSTs will go away when you quilt it. I think you will make your situation more difficult if you use iron on interfacing because you will be stabilizing the bias rather than controlling it.

I would recommend stay stitching from my days of making clothes. Just sew along the edge within the 1/4 inch seam allowance with a slightly shorter stitch.