I am debating between getting a Axiom streamliner DLX , which I think mounts behind your axle somehow, or a more typical axiom rack that mounts to eyelets. Is the eyelet one going to be more stable?

I would expect so.

Not sure what you mean by stable. If you're talking about the load, the top of the streamliner is narrower than most racks so you do't have a very wide platform to put a load on. If you're talking using a single pannier, then the pannier will be located closer to the center of the bike so the bike will be theoritically more stable. If you're talking about reduced weight capacity due to the setback, the manufacturer will state the capacity so you can evaluate wether it suits purposes.

With the streamliner one thing to be aware of if you have a bike with track (rear horizontal) dropouts, is depending on where the rear mounting holes are located the brackets that offset the rack can obstruct the dropouts making it impossible to get your wheel off without taking the rack off. (happenend to me so I fabricated differently shaped brackets with some stock from home depot)

May I say that I've installed about 5-6 different on all the bikes... covering a number of different makes and models. I haven't noticed that one is particularly better than the other. They all mount to eyelets and stability has never been an issue. A couple of years ago I bought the "Bor Yueh" $12 rack from Nashbar for two bikes. Seemed pretty reasonable to me.

If you've got eyelets for a rack, by all means you should get a rack that mounts to the eyelets. If you don't have eyelets, the Streamliner DLX looks like a good option, but it would make wheel removal a bit more involved, and if you've got eyelets it seems like it wouldn't have any advantages.

Doesn't seem as secure to me.

https://www.axiomgear.com/img/prod/st...road-dlx-2.jpg