Generally speaking it is no problem to rent a car at Basel airport. As you can see on the homepage of the airport, you can rent car from at least 6 major and worldwide-known companies.
It isn't a problem to drive a car with your US license, you should just provide a credit card as usual in the whole word.
The price really depends on various factors, as your age, the car category, the duration, the season etc, but you can expect to pay not less than 60 Swiss franks for a day. To quote Lonely Planet:
Car rental is expensive, especially if hiring from a multinational
firm. It's cheaper to book ahead from your own country, but you're
still looking at Sfr350 to Sfr500 per week. The minimum rental age is
usually 25, but falls to 20 with some local firms, and you will always
need a credit card. It is possible to drive Swiss hire-cars into most
EU countries, including the 10 member states that joined in 2004.
The road conditions in Switzerland are excellent. Just beware of speeding because speeding ticket prices are enormous! For example I paid once 30 franks for 1 km/h above the limit. The speeding ticket prices are unlimited. Additionally, if you go to more remote areas you should expect very narrow and windy roads that can be covered with snow and ice in winter.
If your destination is in the city center you should be aware that parking can be very expensive in city centers. You pay there up to 5 franks for one hour. Lonely Planet has some additional tips:
Street parking in the centre (assuming traffic isn't banned, as it
often is) is controlled by parking meters during working hours (8am to
7pm Monday to Saturday). Parking costs around Sfr1 to Sfr1.50 per
hour, with maximum time limits from 30 minutes to two hours. Central
streets outside these metered areas are usually marked as blue zones,
allowing a 1½-hour stay during working hours, or as (increasingly
rare) red zones, with a 15-hour maximum. In either of the latter two
cases, you need to display a parking disc in your window indicating
the time you first parked. Discs are available for free from tourist
offices, car-rental companies and police stations.
On the other hand, public transport in Switzerland is very reliable and especially in city centers very frequent. But it is also very expensive. So it really depends if your destination you have to go day to day is also in the city center or far away. If it is in the city itself, Andra's answer is very useful. If it is not in the city center, you have to consider how much a train ticket will cost. You can expect to pay around 30 Swiss franks per 100 kilometer for long distance trains (in second class, first class is approximately 50% more expensive). I really recommend to buy a ticket, because ticket inspectors are frequent, and traveling without a valid ticket costs 80 franks for the first time, 120 for the second time and unlimited for the third time.
If you travel a lot you should consider to buy a travel pass called Halbtax. It costs 120 franks and is valid for one year. If you have that, every ticket prices is only 50% of the original price. This can be economic even if you stay in Switzerland for only a few days.
To tell you which option is cheaper, we would need to know your exact journey.