So the obvious use of this is to use an outer Barrier to surf into a server, and this probably IS the strongest use of the card. However it says before OR after, which means you can surf a Barrier forwards OR backwards if you want. Why would you want to head OUT of the server to re-encounter ICE you've already hit? There's one main reason: positional ICE.
That's right, the classic Shaper trick of reorganizing the defenses so Chum is on the inside. Now available to Anarchs! The position shift is permanent (same unfortunate reason that unless you're packing Paintbrush or other ICE subtype tweaking you can only use a given Barrier to surf in once), so once you've backed their ICE out past Chum that two-deep server suddenly looks a whole lot funnier. Pachinko outside Data Raven! Counters Sensei if anyone actually used that! Farm the Little Engine for credits...maybe...? I don't think that last one is possible outside of stupid numbers of Parasite counters (especially given that this costs two creds to activate), but it would be pretty funny to watch.
Of course, these are niche uses. The real power of this card is surfing Curtain Wall all the way into a glacier, then breaking it with D4v1d. And, of course, importing this into Shaper for use with Paintbrush and Tinkering in a deck already built to abuse those. Or Escher, which can either let you in everywhere once or create a Barrier-dense server that can then just be surfed over every run. Again, this shouldn't be the ONLY reason you're running Escher, but if you happen to stumble across both (haha, what's this SMC doing here?) it can be entirely possible to close out a game by just making everything surfable. But even without Shaper synergy, Surfer can get you into servers you have absolutely no business being able to access.