When you use "at", you need to remember there is some kind of transit. "At" is a point which you meet. So if you use "at the border", it means you have travelled there and reached that point.

It's worth knowing the different usages of "on" and "at". If you are "at" the border between two countries, you are in the general area of the border. You could be waiting to go through customs, for example. However, if you are "on" the border, you are physically on it, probably standing on it. In some places, you can stand with one leg either side of the line, "on" the border. When speaking figuratively, it is usual to say "on" and not "at". We don't usually say "on the border of" but use synonyms like "edge", "verge" and "brink". I don't know why, but we don't usually say : "He is on the border of e.g. madness. But it is common to say : "He is bordering on e.g. madness".

Preposition-wise it is OK, I think. 'on' would also work. It doesn't quite sound right to me though. Perhaps 'on the verge' would be better?