There is indeed a quality loss if you do changes to a jpeg picture. So if the border is really thin and the image already jpeg, it is better to keep it if the picture.
Of course it is better to remove it before saving to jpeg, but for contributions already received we can't do much except asking to be more careful next time.
It is good to avoid even thin borders, since (as I explained elsewhere) due to jpeg algorithm the quality of the image is bad near sharp lines. So a thin black border produces artifacts in the zones around that border.
Horizontal or vertical, that does not change: borders should be removed