Borders league of legends

The Season 2 borders weren't awarded until the end of Season 2. You won't see anything from Season 3 until the end of Season 3. You will need to be silver rated (or higher) at the end of the season to qualify for the prettier stuff.

Also note they might decide to do something different at the end of season 3.

League will always display your new border.

At the end of every season your old border will be "deleted" and replaced with the new one. Even if you haven't played a single ranked game in the new season your old boarder will be gone.

as pointed out by peacedog, keep in mind that the new boarder will be awarded after the season ended