Ahh! So now we have the real argument -- we need a minimum wage to prevent other laws from screwing things up. There is no screwed up regulation that can't be fixed by more regulation of something else, right?
No. We need a minimum wage so long as there's
any kind of safety net to stop people starving en-masse in the street, be it government-provided or charitable or even just assistance from more well-off relatives and friends. Otherwise employers' wage costs will end up getting subsidised by that safety net - they'll be able to employ workers for less than it costs to keep them healthy and fit for work - which is going to lead to some, errm, interesting distortions in the market for labour. You might be able to "solve" this by prohibiting anyone from providing this kind of support to anyone else, but this has the issues that it's unenforcable and will lead to people starving en-masse in the streets.
As the somewhat more rational supporters of the minimum wage noted, the only people who would have even a chance of "racing to the bottom" would be those already working minimum wage, and most of THEM are either working very simple jobs that pay poorly regardless or are using the job to gain experience for a springboard to a better job.
While the race to the bottom would indeed screw over lots of people currently working for the minimum wage, they're not the only ones it could affect. Suppose you're working 40 hours a week at the minimum wage of $7.25 a week, which is just enough for you to survive on, and when the minimum wage is removed your employer looks at all the unemployed people out there and decides he could get away with paying you $6 a week - not quite enough to make ends meet. What do you do? You get a second job - say, for 10 hours a week at $4/hour. Except that everyone else is doing the same too, and the resulting increase in labour supply pushes wages down further until you're now working 80 hours a week at $3.50/hour. (That's the optimistic version - if the amount of employment doesn't keep up, some people will end up working 0 hours a week for $0/hour as they get replaced by better employees that are suddenly desperate for more work.)
Of course - and this is the fun bit - said increase in labour supply makes it harder for the unemployed to get a job, counteracting the benefits of increasing the number of jobs available. In fact, removing the minimum wage may well make finding a job harder, not easier!