I am trying to downsize some transparents images in PHP with GD, and whenever I do, there is a weird black-ish border that is added around it.
Before
I think the problem here is your source image.
What you have is not a true-color PNG with alpha channel, but an indexed-color PNG with a transparent color. This is apparent if you open the image in Photoshop:
This image was created with anti-aliasing already (which gives the yellow text that white-ish border seen here), but when you re-size it, the sub-pixel calculations may go outside of their borders a bit.
I suspect if you fix the image, making it full RGB with an alpha channel, you won't have this problem.