Well, your current ranks in flex and solo/duo don't matter, as your border(s) are from the previous season. Technically this seasons rewards haven't been announced yet, but it'll probably be pretty similar to whats' below for how border work this season based on your ranks from last season.
(From this preseason's FAQ)
Loading Screen Borders
Part of your season rewards package is a loading screen border. Your new border will be visible once you receive your ranked rewards. There are a few new queues this season, so here’s where your borders will appear:
Ranked Solo/Duo - 2016 Dynamic Queue
Ranked Flex 5v5 - Best of 2016 Dynamic Queue or Ranked 5s (not available until Patch 6.23)
Ranked Flex 3v3 - 2016 Ranked 3s (not available until Patch 6.23)
Normal Summoner's Rift - Best of 2016 Dynamic Queue or Ranked 5s
Normal Twisted Treeline - Best of 2016 Dynamic Queue or Ranked 3s
Co-op vs. AI Summoner's Rift - Best of 2016 Dynamic Queue or Ranked 5s
Co-op vs. AI Twisted Treeline - Best of 2016 Dynamic Queue or Ranked 3s
Custom - Best of any 2016 queue
Rotating Game Modes - Best of any 2016 queue
ARAM 5v5 - Best of any 2016 queue