Stock market winnings are taxed. It’-s called capital gains, and the rate for short term trading of stocks is the normal income rate for individuals.
Of course, this tax will be felt for ordinary investors that hold mutual funds. If the average mutual fund has 100% annual turnover, a 1% tax becomes 2% in additional fees per year, or almost 50% over 20 years. You are arguing for making retirement more difficult.
Most fundamentally though, high frequency trading did not cause the crisis. If anything there was too little, not too much, pricing of mortgage securities. Think about it: even if high frequency trading increases daily volatility, that is not the kind of volatility people care about. People care about booms and busts in asset prices. They care about volatility at the monthly and annual level, which short term trading naturally has less to do with. Higher frequency traders buy AND sell.
Do not confuse flash trading with high frequency trading. Do not confuse problems related “-short-termism”- incentive effects with short term trading. Do not confuse volatility over short time frames with longer term volatility that actually affects peoples’- lives.
A transaction tax might reduce volatility over short time frames, but it will probably increase the serial correlation of markets —- their momentum —- which may increase volatility at longer, more relevant, time frames. If trading is more expensive, booms and busts will be more prolonged.