Aesthetically, it is all about the image. That said, often times a borderless photo has been trimmed down from a fuller image by the newspaper or magazine from whose archive it was pulled. In that situation I prefer to have a full photo. I will take the trimmed one if I really want the image but all things equal I'd prefer the full photo.
I mean, I had to have this one of Jackie sliding into 1st against Chuck Connors:
And the partial border was not a deterrent, but I sure would have liked a complete border.
But then you get into wire photos, and those are all over the place in terms of borders:
