I'm more in line with Red's line of thinking here. The elements wouldn't balance anything of course, they're an added dimension to gameplay after all. But what I'm going with the most is the idea that the best form to implement this would be by having base elements and ways to change those elements later on.
In fact, I dare say, aside from obvious exceptions such as fire rain or water balloon, I'd make the basic element of all rings be "neutral", which is to say, the rings would work as they do now.
Another thing I would do is to prevent rings from having more than, say, 3 alternate elements besides the "base" one. And never less than 2. That would make certain rings just not too good to use if you're aiming to have a major advantage in an area - also, the need to keep the base element as neutral comes in here again so that people who just don't want to care about elements or really want to take a specific ring can still bring whatever they want without being hindered.
Animated would
need a preset pool of possible resistances they can have, at the very least.
Making it random would only cause players to feel compelled to bring as many elements as they can, in a random mesh, which would beat the entire purpose of the thing. Players, even "hardcore" ones would certainly learn to see past the system and run "freelancer roles" constantly, as they do now, and its implementation would have been for naught.
Making it static would... Make the game too predictable and too easy screw with, as everyone knew what was in store.
Best way would be to make the elements come in play as you progress or decide to challenge yourself. easy modes should have predictable weaknesses with no resistances, like, monsters getting 100% of a elemental armor type (All saws are Flame Weak and Earth resistant); while Hard mode should mix 3 types of enemies with weaknesses that are somewhat covered by the rest of the mob (Some are weak to Fire or Demonic or Earth. Others are resistant to Fire or Earth or Water). Extreme mode would have Resistances to even the "neutral" type, and element absorption. (Saws are resistant to Neutral, Earth, Fire and are weak to a random element that might be one of the resistances (Ignoring the fact it was a resistance). The randomness is applied to all saws in a wave rather than individually)
Elements also add a cool possibility to the game. Let's say that all enemies have "hidden" elemental resistances. Let's say that there's a ring that for 8 stamina, let's
you and your whole crew see a symbol over the enemy's portrait that tells the kind of "elemental armor" the enemy has. That's a new non-offensive, non-defensive, non-CC'ish but still essential rings to be had one copy of in any good crew.