The same question was just ringing through my head because I'm also a day trader, but the only thing that I could think of right now is that day traders can reduce the risk by minimising the amount they leave in the exchange on a daily basis since there will also be open trade waiting for the right order to hit in order for it to execute.
I also think that spot day traders can make use of alarm. But this is not convenient like having the coins on the exchange. To be moving the coins from exchange and wallet, stable coins like HSDT and USDC has low fee. It is even zero fee on some network in some exchanges. Also very low on some network like Optimism. Making use of lightning network for BTC if the amount is not huge.
Day traders can't completely avoid leaving some money on their exchange account even if they can buy tokens and move out to their private wallet. How about those that trade on derivatives?
If not use high leverage, you may decide not to leave all the coins on an exchange. You want to trade with $100 with 1x leverage, you can go for $20 with 5x leverage for bitcoin, not altcoins that are very volatile. But I think you can go for 2.5x for some altcoins. So deposit $20 for bitcoin. $40 for altcoins. But this are people that do not want to make use of high leverage.