The concept of being a race to the bottom is just a flat out fallacy. It's all theory, but in reality it doesn't happen. Unless of course, by bottom you mean technology becoming so cheap that even the poor can easily afford it.
So far we've seen the following implemented under the idea of prevention of the so called race to the bottom:
- Ludditism (i.e. banning technology because in theory it takes away jobs)
- Lump of labor (mandating reduced working hours to increase employment)
- Living wage (aka minimum wage)
The first three work under the assumption that the demand for labor always remains constant. That alone is a huge huge mistake.
As for the first one, even if new technology doesn't create new jobs while getting rid of the old ones but in the short term results in layoffs, in the long run the demand for labor continues to rise. So while your job might go the way of the dodo, new jobs will spring up somewhere else, likely in another unrelated industry. For example, in the classic case of the original Luddites, when it became cheaper to make clothing due to technological advance, people had more money to spend elsewhere, and that elsewhere is where the new jobs go, and you invariably see more jobs created than had existed before that "job costing" technology was invented. It happens every single time, without fail. No matter how automated things become, people will always find some place to spend their excess money.
As for the second - the argument that we need to reduce working hours is often tried, and instead of getting reduced unemployment, you get the opposite. Every single time. Counter-intuitive I realize, but it is a fact nonetheless. The reasoning for this is rather simple:
As for minimum wage...If your wage was below what you're willing to work for, then you'd simply not take the job. This is a fact. Most people do in fact work at above minimum wage. Minimum wage has the following effect: For the low end workers who really aren't worth a shit (there are many out there - this nobody can deny) they simply have no job at all. Whereas they could have at least had SOMETHING, they now have nothing. This has two effects: Increased unemployment, and higher prices. Because prices now go up to match that minimum wage increase, your purchasing power hasn't really gone anywhere. Most people tend to equate money with wealth, and that relationship isn't one to one.
A classic example I look at is this: Back in 2001, I paid $3800 for a 50" tv. The thing was pretty massive not just in diagonal length, but it was pretty fat too - it was rear projection. A really big and obnoxious TV by today's standard. Last year I "upgraded" my whole living room: Bought a new leather couch, ($1,200) built a 5.1 surround system from the ground up (none of the HTIB crap, a truly good sound system, $1,000ish with 8 channel lossless audio) and a new samsung 50" tv that has a MUCH better picture quality than the one from 11 years ago, consumes a lot less power and is light enough for one person to carry. Total spent was $3,200. Basically by spending less I have more "wealth" than I did 10 years ago - and that's even ignoring inflation.
When somebody tells you that the poor are poorer and the wealthy are wealthier because - adjusted for inflation - the poor are making fewer dollars today than they were in the 90's, they have no idea what they are talking about. Even though they say wealth, they aren't looking at wealth, rather they are simply using the medium of exchange as a guide. In reality it doesn't work that way. The example I gave above applies to a lot of things beyond just entertainment. Cheaper shoes, cheaper food, cheaper you name it. Something I might add - I have less money today than I did in the 90's. Mainly due to economic issues I am by the liberal definition "poorer", but nonetheless I have it better than I did then - I have a better car, I have a better house, I have a 50mbit internet connection at a lower cost than I used to pay for dialup, and I have a better entertainment system. But, according to the liberal, I am poorer thanks to the wealthy being too greedy, so clearly we need to raise taxes.
By the way, even the most liberal of liberal economists will tell you the same thing. The champion economist of the democratic party nearly all of the left politicians model their ideals after (including Obama,) Paul Krugman himself believes that sweat shop labor is better than no labor, and minimum wage causes more problems than it solves, and ultimately makes things worse. Of course where they differ vs say chicago school or austrian school economists (which the right and libertarians tend to follow) is with regard to how the government spends. But nearly all of them agree that there shouldn't be wage floors.
Now as for tariffs - many believe that because one country can make things cheaper than another country, then clearly we need tariffs to balance things out, otherwise we're just racing to the bottom. Not so fast. The reason tariffs are a very bad thing is because they completely ignore a concept economists refer to as comparative advantage. The original keynesians (basically Keynes himself) said that tariffs just do the trick. The new keynesians (such as Krugman - nobody follows old keynesian theory by the way - it was disproven in the 80's when it couldn't explain stagflation) will tell you that the only thing they do is make things worse.
The chinese for example, have a much better/more efficient capability than we do to create textiles and assemble electronics. We on the other hand have much better capability of fabricating semiconductors and building capital goods like jumbo jets and earth movers (by the way, in spite of what most people think, manufacturing is still big in the US - in fact we're #1 - whereas most people are lead to believe that we don't really manufacture anything and are just a service economy.) The best thing we could do is let china create the shirts, and us create the earth movers, and then trade them. This results in increased wealth for both countries at no net expense to either one. For example, by buying cheap shirts made in china, I can now spend that money saved elsewhere, meaning I have more wealth. The reverse is true in china - although they make earth movers, theirs are either vastly inferior or cost more. So, ditch the tariffs. Smoot-Hawley is a great example of the destruction that tariffs cause - in fact tariffs were THE cause of the great depression.
In the end, I'm not sure what part of "race to the bottom" you think is so bad, but the reality is that there hasn't been any harm caused by this so called race, rather it has only resulted in things being better for everybody, including the poor.