I've been living in Basel for eight months now. Here's what I recommend for you.
We always take our visitors first to the Old Town of Basel - where the cobblestone streets are narrow and architecture is extra charming. The walk is filled with small boutiques selling scarves or gloves or trains or shoes. Or buttons. There is something for everyone.
Along the way, we stop at Gilgen for pastries and/or a coffee at the Mitte, which is very child friendly. You know you're in Switzerland when a child-friendly cafe has five Eames chairs for kids to swirl around on.
Start at the top of the Spalenberg (in the Old Town) and work your way down.
You can't miss the Muenster. The view of Kleinbasel (Little Basel) from the Muenster Church is beautiful. There are five bridges in Basel - Die Mittlere Brücke (The Middle Bridge) is the most famous to walk across - though my favorite is the Birsfelden dam (technically not a bridge). All bridges are bike and pedestrian friendly.
It's true that in the summer people spend their time floating down the Rhine. Georg and I did it this past summer and it was awesome. Do it if you can. On the Little Basel side, stop at Pulpo for a beer afterwards. It's also a great spot for kids - it has a room with sweet Swiss toys to play with. Or on the Big Basel side, go to the beer garden - Veranda Pellicano. They have standard hamburger/hot dog faire and grilles for doing your own grilling. Best of all you can sip beer and watch your kids throw rocks into the Rhine nearby.
As everyone knows, restaurants in Switzerland are expensive. But one needs to eat! For a casual and friendly experience, our favorite is Consum. This is a great place at any time really - an early glass of wine, light tapas, or late into the evening shenanigans. Not great for kids.
If you do find yourselves wanting a more fancy dinner out - we've gone to Goldenes Fasse a few times and it's always been delicious and fun. Not a kid place.
A good spot for kids is the MarktHalle - it's an indoor market with plenty of space for kids to run around and even has an indoor slide. The food is good too! Best for evenings or weekend brunches.
For drinks, I would go to Consum (above) and also Cargo Bar. Cargo Bar is on the Rhine, and has a younger crowd than Consum. It's the kind of place I would have spent many long nights in my 20s.
Here are a few spots I like and would highly recommend making your way to:
Hasenburg - very local, Swiss restaurant and bar. Open all the time. Fine for kids but more fun without.
Grimsel- high-end Swiss design store
Spielegge - toy store with classic Swiss toys
Baeckerei Kult - bakery with sammy's!
St Johanns Park & Cafe - cafe at St. Johanns Park