Are forum badges worth anything?

No, nothing and they are pointless and in a lot of cases simple to get.

And they are obviously most commonly known from Steam, where again outside of personal satisfaction they shouldn't really mean anything to anyone but yourself.

Not sure if anyone has academically studied this but it will play on the compulsion built into humans to achieve/advance, even when it means nothing, and could be used to direct their behaviour.

Ingame achievements work like that on me, I tailor my game playing to getting achievements and when I have got all the ones I want and finished the game I commonly lose interest.

Here it isn't anything more than validating engagement, but again Steam points the way of how it can be used for revenue generation.

Steam levels - you get 'higher' levels the more money you spend as it is based on 'experience' gained primarily through buying Steam trading cards, which have no purpose other than to generate badges that again are purely aesthetic.

On top of that the badges have levels and also foil, so you can level it up and also get a - very expensive - foil badge.  Steam gets a 2-3 cent cut on average PER card transaction, so they want millions of transaction to be being made regularly, they don't have to do anything to generate large amounts of income.

I am level 44 since they were launched with a 15 year account, I get XP for doing the community things in sales and crafting badges that I only buy the cards through selling the free to get event cards on the market.  I don't spend any real world money as it is self perpetuating, but I have in the past and someone has to to buy my cards.

I have 769 games and 74 badges.

The highest level is 4575 for a 9 year old account and the account is 'valued' at $8,512-$36,341 which is the price/on sale range that the games could have been bought at.  The player has only played 111 of their 4,719 games.

The third highest account is only registered 1.7 years ago, so it is possible to buy your way to the top as it is designed to encourage.

TLDR - the badges mean nothing here and cost nothing but there is a lot of gambling/engagement/addiction psychology that goes into these sorts of things that can encourage certain people to spend A LOT of money to achieve them in places where they are being used for that purpose.