Cover the entire background of your crosshair with pure black at alpha 1, so that you can't really see it at all but it's still there. This has always solved all my issues with white borders from blending alpha areas, imo is the easiest way, and doesn't require messing with the black border itself like Enderp's method.
I was going to mention the Point Sample vtf flag too, so I'll just elaborate on Enderp's description. Point Sample will make it completely pixelated--resizing doesn't blend colors any. So this should solve the issue of the white border when resized, and if it's a good thing it'd make the black and the yellow of your crosshair stop mixing together as it gets smaller. The issue with it though is that if you resize it to be small enough, it loses pixels and some sides might be smaller (and in the case of having black borders, you might start missing them on some sides). My opinion for Point Sample is that it's good for when a hud element is slightly resized because of resolutions, but not much else.