Swiss wine towns near Basel

I don’t know of Swiss wine towns near Basel. They are mostly south and west. The closest may be near Murten, which is close to Bern.

There aren't any Swiss wine towns near Basel. There is a small amount of specialist Swiss vineyards in the vicinity of Basel, most Swiss wine comes from further south, Valais for example.
There are plenty of wine villages near Basel, If you go to the look-out by Basel cathedral, you can even see some vineyards in the distance, a few inside but mostly just outside Basel city limits, within walking distance of a Basel tram. These are however in Germany.

Website with signposted walks through vineyards on outskirts of Basel: https://www.derweinweg.de/weinweg/tourenportal/

The "Badischer Weinstrasse" starts on the edge of Basel. This is the German equivalent of the Route du Vin d'Alsace". Every 2nd or third village heading north out of Basel towards Freiburg has a "Winzergenossenschaft" (Co-operative winery, jointly owned by all farmers in the village, equivalent of a French "Cave co-operative")
These vineyards are on the lower slopes of the edge of the Black Forest, facing across to the Alsace vineyards on the lower slopes of the Vosges on the opposite side of the Alsace plain but further north.

The local speciality is "Spätburgunder", a red wine.

Badischer Weinstrasse website (German only): https://www.badische-weinstrasse.de/
Click on "Karte" and you will get a zoomable map of the wine sites.
The first two Winzergenossenschaft north of Basel are:
https://www.wg-haltingen.de/
https://www.markgraeflerwinzer.de/
The first photo on both those sites show the view from the vineyards back towards Basel.

If you have a car, head north out of Basel via Otterbach Grenze (Border post) onto the German B3 - this is the old main road to Freiburg through all the villages, NOT the Autobahn. You will soon encounter vineyards, restaurants and Winzergenossenschaft.