I read a few of the other answers, and took notice that none of the other users come from the region.
Since I have been living in basel for more than 20 years I will give you detailed spots on where you can take great photographs given that you have a limited time frame.
Basel is a very touristic town with many people roaming the streets on a good and sunny day. The people are friendly and I have had mostly success when asking if I can take a portrait. You will have daylight up until 9 o'clock in the summer. I have been doing several projects in the area of portrait like the 100 strangers portrait challenge.
Here are various Hotspots in the center of town with a link to googlemaps:
This is a very long street with a broad variety of photographic opportunities.
Along the rhine especially in the evening you will find people enjoying a drink, cooking meat on a grill,swimming in the rhine (from july-august), and maybe even jumping from the mittlere Brücke. All in all the people are very pleasant especially after work. The best times to take photographs is between 5-9pm.
This is a similar good spot like the freiestrasse. People will go out after a hard day of work and enjoy the evening.
To sum things up, those three places offer great opportunies for portrait and/or street photography. You will have people from many different cultures in a relaxed and great ambience. Of course there are a lot of other places to take pictures, but if you have a tight time frame, then I suggest that you walk from one of these spots to an other.
Without leaving basel, there are some great places from where you can take pictures of the rhine during sunset.
From this spot you have two options. During the day you can go all the way up on top of the münster church, from there you will have a 360° view of Basel.
Opening hours during summer time:
Monday to Friday: 10.00-17.00
Sunday and festivities: 11.30-17.00
An other option, If you walk behind the Münster you will have a great view of the rhine. From there you can capture stunning pictures of the rhine banks and the "mittlere Brücke"
You don't have to go in the Münster, you can also walk around it.
I hope you have a great stay in basel and will benefit from these photo opportunities. Maybe I might even see you walking down the streets avidly taking portraits.