Basel is small, you can get anywhere to/from anywhere easily by bike or tram/bus. Without the complications of kids, walking is also usually possible from anywhere to anywhere.
I live in Brausebad. I like this area as it's at the intersection of trams 1 & 6, next to a Bus 50 stop (main station in one direction, airport in the other) and also a park.
It's about 15 minutes' walk to the town centre.
However, if you need a bigger apartment (three bedrooms and up) they're easier to find and usually a bit cheaper on the other side of the river (area called Kleinbasel). Also, there, is a relatively well stocked Asian supermarket as you'll need to cook for yourself regularly (you'll learn what I mean once you've tried the Asian restaurants in Basel). However, as before, that supoermarket is easily accessible from anywhere in Basel really.
I talk about (Asian) food a lot. If I've been at all helpful, and you can cook, there's an easy way to thank me