Do day traders – particularly those who trade futures contracts for minutes at a time – serve any useful purpose in our economy?
Yes. Otherwise, they wouldn’t make money doing it.
Make day trading illegal and see what happens to the stock markets and capital markets in a country. 
I think Art Carden touches on this in his Short Selling: Explanation and Defense lecture.
This answer is not sufficient proof. Thieves make money too, that doesn’t prove they serve a useful purpose.
Please try again.
The traders aren’t stealing or breaking laws.
Define “useful purpose”
Good point. “Useful purpose” is pretty subjective.
If I had to say they did something “objectively useful” it would be that they help in the catallactic process and they help to factor in information into the market price of things.
I mean they might make themselves some money and then go spend it on something or maybe save it or maybe invest it, those can all be “useful” things.
Do appraisers serve any useful purpose?
The other thing that we should probably define is “day trader”
first of all, the name is a little misleading and second of all because not all day traders trade and/or speculate on the same things. There are day traders that trade stocks, derivatives, CDSs, commodities, futures, god knows what else.
No, they just throw dices and act accordingly.
FYI:
I just listened to the Art Cardin lecture. The bit about day traders was near the end, answering a question from a student. He only said a few sentences:
“Day traders produce information, they’re helping to align the prices of assets to what they would be if everyone had perfect information.”
“they’re providing liquidity”
Yeah right
From Wiki:
Day trading refers to the practice of buying and selling financial instruments within the same trading day such that all positions are usually closed before the market close for the trading day. Traders that participate in day trading are called active traders or day traders.
Some of the more commonly day-traded financial instruments are stocks, stock options, currencies, and a host of futures contracts such as equity index futures, interest rate futures, and commodity futures.
Day trading used to be the preserve of financial firms and professional investors and speculators. Many day traders are bank or investment firm employees working as specialists in equity investment and fund management. However, with the advent of electronic trading and margin trading, day trading has become increasingly popular among casual, at home traders.
Types of day traders.
Also:
Because of the nature of financial leverage and the rapid returns that are possible, day trading can be either extremely profitable or extremely unprofitable, and high-risk profile traders can generate either huge percentage returns or huge percentage losses. Some day traders manage to earn millions per year solely by day trading.
Because of the high profits (and losses) that day trading makes possible, these traders are sometimes portrayed as “bandits” or “gamblers” by other investors. Some individuals, however, make a consistent living from day trading.
Ok that’s a decent definition I suppose. I guess there could be two day traders or three day traders or week traders. I suppose that’s kind of irrelevant though.
Yeah, like any other speculator, they buy low and sell high, thereby balancing out prices of stocks during the day.
What would happen if they were not there to balance out prices during the day? Would their absence cause any harm to our economy? (Wouldn’t prices balance even if they weren’t making their 5 minute trades?)
Do day traders add any value to our society? (Or would we be better off if they all quit their “job” and started producing goods and services instead?)
They provide capital for minutes at a time.
Without day traders, intraday prices would be much more volatile and they would swing violently based on the purchase or sale of a few investors. Would this be bad? Sure. Would it be horrible? No.
Easy answer - they provide liquidity, which allows other, larger market participants to execute their orders at better prices. You don’t have to look any further than liquidity. A liquid market is a good market. You want to get the best price possible no matter what transaction you are trying to make happen. The less volume and liquidity, the more likely you are to get filled at a worse price.