My advice would be to actually drive in the city. Getting into the area of Heuwaage is your best bet as it's easy in and out on the highway to the airport and has a big enough parking (it's less than 10 km about 15 min drive). From there you have either the option of Markthalle (2-3 min. walk) with various very good "street food" options and a fantastic beer tap place. Alternatively, you can walk into Paddy Reillys which is literally at the exit of the parking for a more traditional Irish Pub experience.
La Diligence, 8 Rue de Mulhouse, 68300 Saint-Louis, France
Au Bœuf Rouge, 9 Rue de Hesingue, 68220 Hégenheim, France
Ye-es, a US bar feel. Absolutely not a chilled pub atmosphere which is what seems to be asked for.
To kill a few hours I'd recommend chilling at the nature reserve of the Petite Camargue Alsacienne
All just a stone's throw from the airport, just the other side of the A35 motorway from the French side airport entrance (second motorway exit going North out of Basel).
Sometimes I contain my hunger and then eat on the plane. Easyjet food is better and more affordable than you might think. Especialy in comparison to what the airport has on offer. Which admittedly isn't setting the bar very high.
Either that or bring a packed lunch.
If you don't want to go into downtown Basel there are also plenty of local restaurants on the outskirts of Basel. A quick google search will reveal some. None of them exceptional AFAIK, but if you want something warm in your stomach and a beer to wash it down and not pay tourist prices, that's what you can get. Many of them easily doable on public transport if you don't want to drive. The airport bus does several suburban stops before getting into downtown Basel.
There is very little of note on the French side AFAIK and not really cheaper either as it's still too close to the border.