I wonder how things would have turned out. I'm going to take them as a total package of ratings, little boxes, powerful PvP gear, and the increased focus on PvP. Maybe that makes the last one the best question, how would things have turned out with less focus on PvP?
With no microscope to PvP balance, we'd have probably seen fewer PvP buffs and nerfs. This would mean weaker balance. As a result of that, BGs would have even worse balance. If you think DKs are bad now, imagine before all the nerfs. Some balancing would have still happened, but I think a lot less and PvE would have been driving more of the balance, at the potential cost of PvE. Strangely, arenas may have saved BGs.
However I cannot say this is absolute. BGs survived for years with terrible balance. They had rewards, but the time cost was high and I'm not even thinking of the rank rewards. TUF was a pretty powerful weapon, comparable to MC weapons and not too much worse than BWL; however it required exalted with AV and that was a long grind. Other BGs offered powerful gear, with WSG having some of the most powerful pants in the game (at least as I remember), but WSG exalted is possibly one of the longest grinds ever. BGs might have persisted with a new level of terrible grinds, though Blizzard would have to figure out a way to not instantly hand out gear because someone hit exalted at 60.
Without the powerful gear given by arenas we'd likely see the continuation of the old PvE to PvP. BGs would be dominated by PvErs, not PvPers. Again, arenas may have saved BGs. Or could updated rep items been strong enough? They likely would not have lasted through the addition of new raid tiers.
But without the surge of PvP gear from arenas and the non-set honor pieces, would we have played differently? Arenas, more than BGs, drive players to improve their gear. Without arenas would people have chased PvP gear as much? Would Blizzard have even added PvP gear? Without the gear motivation, BGs would exist only for PvP, not for gear collection. But we'd also see fewer people, since they tend to follow the gear. Arenas saved and/or killed BGs.
I benefited and/or was indirectly hurt by arena gear. The pieces from it allowed me to get T6 equivalent despite having no chance at all at BT. While some stats were wasted on resilience and excess stam, arenas allowed me to reach a much higher level of gear than I could have otherwise. This saved me from needing to compete for T4 and allowed me much higher damage than I'd have had otherwise. However it also made me care much less about early raids, it removed an incentive to raid.
Could arenas have hurt or helped raiding? They did provide an alternate and often easier source of gear. However that gear might have been beneficial, allowing a player to effectively get the gear from a boss before killing it, allowing for faster and further progression.
What do you think would have happened without arenas?