After spending several hours researching, there still appears to be little consensus on this topic. But here is a summary of potential solutions.

The Problem

But first, the problem: the issue with High Efficiency Front loaders is that the vibrations are vertical. The drum spins horizontally so the vibration is all in an up-and-down motion. It's also incredibly high speed. The problem is that wood framed houses aren't designed to withstand the vertical vibrations of these newer washers that well. a HE top loader, while also fast, transfers the vibrations horizontally, which the floor framing of a house is better suited to handle.

The List of Possible Remedies

Get a top-loader instead.

A traditional top loader (agitator) spins, but not nearly at the same speed as a front loader. It also does it horizontally so doesn't transfer nearly the vibrations that a HE FL will.

The catch is that these are harder to find and more and more top loaders are now HE top loaders. Instead of an agitator, it uses jets of water. Reviews of HE top loaders seem to be mixed. However, there seems to be consensus that they don't clean quite as well as front loaders (a con) but that they also don't vibrate as violently as a front loader (a pro if vibration is an issue in your house).

Only put HE Front Loader washers on top of concrete slabs.

10 years ago, this was supposedly the only option. Apparently technology has improved to allow these to be installed on wood framing--however many professionals still do not recommend installing it on a second floor (as second floors tend to not be built to handle the same loads as the first floor).

Make sure the washer's feet are all level.

This is the easiest/first thing to check. I thought mine were but on careful examination I was able to improve things further by adjusting the feet. The easiest way to check this is to reach to the back of each rear corner and try shaking it towards the opposite front corner. You're are testing the diagonals to figure out if one of the corners needs to be raised. I had one ever-so-slightly out and adjusting it helped quite a bit (I suggest grabbing the rear corner is that is likely sheet metal and rigid rather than the front, which is often plastic and hard to 'feel' for wobble).

Move the appliance towards a wall/corner

The closer to a corner, the more rigid the structure may be (might be closer to a support beam, walls may add lateral rigity, etc.) I moved ours closer to the corner of the laundry closet and it seems to have helped.

Use rubber vibration feet/pads/matts

There are dozens upon dozens of companies selling anti-vibration feet. A majority of people claim that these help, but a strong majority claim it makes things worse. My (non-physics-expert-mind's) opinion is that when they do work, though you may reduce the transfer of vibration to the structure, you are likely increasing the total amount of vibration the appliance, itself, now has to absorb (reducing its lifespan).

Use pool noodles on the sides

A common youtube/facebook suggestion is to take the pool noodle foam floaties and cram one between each side of the washer. This will only work, of course, if your washer sits close enough to the wall on one side and the dryer on the other. I haven't tried this yet. My (again, non-physics-expert-mind's) theory is that this spreads the vibration load out to more planes/points of contact.

Reinforce the floor

I see many suggestions of this, but not necessarily any first-hand reports as to how succesful it is. I'm guessing it's just because it's likely a pain to deal with. The suggestions are to tear out the floor and then glue-and-screw two layers of subfloor together with a tight screw schedule. The subfloor being plywood or dedicated subfloor products.

In addition, add blocking to the joists underneath and, in extreme cases, ass a support beam/post directly under the laundry area.

My conclusion

As previously mentioned, double checking the level of the feet has helped considerably. And while I'm happy with the improvements, it still vibrates hard enough on its final spin cycle that you physically feel uncomfortable standing in front of it.

I'm now narrowing it down to two options:

1. Try a HE Top Loader. The problem is that I really do like the front loader's performance. It uses less water and cleans better that the top loaders apparently due. And because of the extreme spin cycle, we save a lot of energy with the dryer. As such, I think I'm going to...

2. ...reinforce the floor. Right now the sub floor is chipboard with hardwood flooring. Neither is all that sturdy. I'm going to take out the hardwood (bonus is that we can use this to patch parts of the kitchen later) and possibly the existing subfloor if it's not too difficult. I'm then going to glue-and-screw down one or two layers of 3/4" plywood (I may use maple ply for the top one and apply poly on top to make it a 'finished' surface). I'm also going to add blocking between the joists.

What we ultimately ended up doing

I bought a 4x6 and created a beam in the crawlspace under the washer/dryer area with two lolly columns. Alas, that still didn't seem to remedy the situation.

So, we gave up. I hired an electrician to wire up a space in the garage and I plumbed a new drain line and vent stack and we moved the entire thing into the garage onto concrete.

And things are good now. I do believe that these HE front loaders are truly only really designed for placing on concrete slabs.

The plus side is now we don't have to try and do laundry in a hallway closet.

Yes, as @TomG says, take the standpipe up - but I'll modify 18" to "as far as the ceiling or the 8 foot limit allows", that's the least expensive first approach, and if it solves it, you are done. Get it right up inbetween the floor joists.

If a plumber was involved in the black Tee on the white PVC pipe at the top of the picture, the licensing board should hear about it. That thing is dead wrong - it's installed sideways (it's the wrong fitting for the location). I'd expect there might be venting issues with that connection, too.

If merely extending the standpipe does not do it, you need to clean up the lower mess to gain some height (and ensure adequate flow.) That might well involve a plumber (or some advanced DIY plumbing) and some expense, but it would not involve extra complexity and electricity use.