Stockinette is an unbalanced stitch.
Stockinette works perfectly for:
stockings,
socks (but you need ribbing or something to keep the cuff from curling),
hats (but you need ribbing or something to keep the edge from curling),
mittens (but you need ribbing or something to keep the wrists from curling),
gloves (but you need ribbing or something to keep the wrists from curling),
pullovers (but you need ribbing or something to keep the wrists, neckline, and bottom edge from curling).
Notice! All of the above examples are items that curl around a body part. None are scarves, blankets, or even cardigans.
Stockinette curlsPERIOD
IF you knit extremely loosely, possibly - just possibly - it will lie flat on its own. Otherwise, it curls from all edges.
Yes, you can try using a non-stockinette stitch border. You can add a crocheted border. However, it doesn't always work. Sometimes the stockinette is so determined to curl, it simply flips the neat, flat border entirely to the other side!!
If the fiber is a natural/non-synthetic product, you can block the curl into submission ... until the next wash, when the blocking must be repeated.
If the fiber is synthetic, you can try 'killing' it. Google for how to block acrylic or killing acrylic. Just beware that the fabric will undergo a permanent change. Once cooked, it's forever. Too much heat is fatal to it.