It's actually more complicated than that. Politics shall intrude.
So, here's the deal. Let's say you buy a bunch of stuff from online retailers that don't currently charge sales tax. Once you start paying those sales taxes, what else happens? States don't operate their budgets the same way that the federal government does. At the federal level, there is no real connection between revenues and outlays, which is why we run deficits. States don't have the same capacity to do that. Most are functionally required to balance their budgets. Without the capacity to run deficits, they are dependent on a combination of primarily income taxes and sales taxes to fund their expenditures. Yes, there are fees and stuff, but it is primarily through income taxes and sales taxes that they operate. They have been forgoing sales taxes because of the move to online sales. Give them the capacity to tax that, and they will. Cue The Beatles. Why doesn't that raise your total cost? Because by taxing sales, there's something else your state doesn't have to tax. Specifically, income. States aren't going to run surpluses because they'd rather give someone a tax cut, particularly if the state is run by Republicans. Cutting income taxes is simply a default political move that appeals to a specific constituency. It's just easier.
So, if sales taxes effectively go up because they are imposed on internet sales, then whose taxes go down, or at least, don't get raised the next time the state has to handle some budgetary mess? That's a matter of state politics. That's a political choice to be made by whoever is in office.
There are some considerations. First, whoever gets the state tax cut benefits more. Who will that be? That depends on who is in charge. Republicans, in most states for the foreseeable future, so that'd be higher income folks. Then again, who buys more tax-free stuff on the internet? I don't know how this balances out. If you were buying a lot of big ticket items, tax-free, online, this hurts you, but only higher income folks can do that, so if the higher income folks get a tax reduction to balance out what is effectively a sales tax increase, I don't know what the net effect is. My point, though, is that this is actually slightly complicated. It could be that middle or lower-income folks benefit more by virtue of not buying a lot of tax-free stuff on the internet, and getting an income tax cut at the state level. Or, it could just be completely idiosyncratic, and dependent on everyone's weird buying habits, combined with how that interacts with state tax law.
In other words, it's just not clear to me how this helps or hurts any one, specific person. You'll notice when you start paying that sales tax, and it will piss you off, but there may be other budgetary effects at the state level to balance that out. That will depend on the state.
Then again, state governments, and politicians in general, pretty much have their heads up their asses anyway, so to suggest that they might handle the revenue increase rationally may be completely ludicrous. "May."
[Facepalm...]
Never mind. Why do I keep trying to apply logic to politics? Hey, look! A windmill! Tilt, tilt!