I have lived in both Basel and recently NYC (East Village and S.W. Harlem/Central Park North) so I have an idea of costs.
Both cities are expensive, but in opposite ways.
In NYC, it is the rent that hits you hard. I found day-to-day expenses (food, eating out, clothes, transportation, etc) fairly reasonably priced - though I realise that NYC is very expensive by US standards.
In Basel on the other hand, you'll find that rent is comparitively inexpensive, and that that apartment you get will be a lot better in quality that what you get in NYC. Where you wallet really feels the pain is on the everyday expenses. Groceries are expensive (particularly meat and fresh produce), and eating out is very expensive. The price of a so-so meal at an OK restaurant in Basel will cost the same as a meal at a very good place in NYC.
Living off that one Swiss salary will make things tight but if your husband can find a similarly paying job, you should be fine.
It's also worth remembering that it IS possible to economise, even in Switz. You can do most grocery shopping over the border in France or Germany. You could get a place in Basel-Land, as opposed to Basel-City, which would mean a small increase in commuting time, but allow you to save a good chunk on rent, income tax and the compulsory health insurance.
I also found that because there is so much to do in NYC compared with Basel, it was FAR easier to fritter away money in the US. I don't have this problem so much in Basel.
I personally think that you would be OK on the one salary here, so long as you economise. The fact there might be a second source of income, should make things a lot easier too.
For what it's worth, I would say that even if it might not be the most financially lucrative decision to live overseas (for a while at least), it will be a very enriching personal experience to have, even if you ultimately decide it's not for you and wind up moving on or going back to the US.
Keep us posted on how things shape up and best of luck!