I have a lot of experience with pansies because they grow well here in Massachusetts, Zone 6. I like that they're among the first plants hardy enough to be outside in the spring, sometimes as early as April. When tended to properly, which involves only a few minor things, they'll bloom into the fall, sometimes until the first frost. They grow easily from seeds too, if you're a person who likes to do some of your own germinating.
You mentioned wind and rain, but I'd like to know if this happened during your hottest months. Pansies are not fans of high heat and humidity, especially if they're in direct sun, and that could easily be part of the problem. If I don't move my pansies out of the direct sun during July and August, they'll lie down, become limp, and stop producing flowers. Once the temperatures get back down below 80° f, they perk back up.
Correct me if I'm wrong, but it looks like you may have them planted in a coco fiber liner. These tend to drain well, sometimes to the point where plants dry out and need more frequent watering. If that's the case, I'd be even more encouraged that, although the weight of the rain may have done this, drainage has likely been sufficient enough to prevent full root rot.
I'm with Bamboo here; there are enough green leaves to make me think at least some are salvageable. It's definitely worth investigating before throwing them out. I too would cut them all back to within an inch or two of the top of your soil. It's not just that the weight is causing them to droop, but any energy left in the roots is being sapped in an effort to help the plant grow. Cutting them back is your best chance to give the roots time to concentrate on absorbing the nutrients from the soil and, essentially, to start from scratch. Then dig down a little and see if you have any viable roots. I suspect you'll see some life under there, especially on the plants with the most green leaves. If so, gently pull apart what you can, and choose which to save. Replace your soil with something fresh and dry, or move them to a different container.
Trimming is important even at their healthiest. Once they've grown to a point where the stem is spindly and hanging over, pinch it back to where the next bud or bloom is. It not only keeps the plant healthier, but it also makes a fuller and prettier shape.
I also agree that they're overcrowded. This happens to me all the time. I want the container to look full, so I put too many plants in, only to find that, four or five weeks later, the roots are fighting for space and flowering slows down.
Much of what I've already discussed is useful for the future. To sum it up:
Don't overcrowd the pot. Either leave a few inches between plants, or split them once they become root bound. They transplant really easily, so if you choose to go that route, it's not a big deal.
Keep them well trimmed. Pinch off mature flowers to encourage new growth. Also, any stems that are brown and brittle should be clipped off right away.
Avoid high heat and humidity. Take them out of the hot sun in your warmest months. If you can't move them, make sure they stay well-watered, and don't worry if they stop producing flowers for a while.
Water them regularly and make sure they have good drainage. The roots are generally very hardy, so they're a lot less picky about water than some other plants, but they do appreciate some drainage. If it's easy to move the container out of the way of an oncoming storm, do that, but, if not, check them after to see if you need to do anything. If they're in a small enough pot, you can even gently tip it and pour off the excess water.
Use a well-balanced potting soil. Obviously you already do that.
I hope you can save some of those plants, and that you have continued success in the future!