Places to stay in Basel

If you would like to be close to the main train station, take a look at Hotel Gaia and Hotel Euler.

In the old town area, we liked Motel One. Fresh and spotless, good rate

Sandy - This hotel is excellent. I have stayed there a few nights while in Basel.

https://www.hotelrochat.ch/

Most of the things you will want to see are in or near the Altstadt, between the main station and the river, so stick to that area. There are several hotels on Centralbahnhofplatz, next to the station, which is convenient for all forms of public transport. The Altstadt streets are hilly in places, so be sure to check the hotel you pick is accessible for you.

They are all listed on the official website: https://www.basel.com/en
You can look up prices and book them there.
Yes, make sure you stay in one in the city centre, or for convenience around the station.
Once you check in you will get a free transport pass (trams+buses) for the duration of your stay.

Around the station: Victoria, Schweizerhof, Gaia.
A few minutes from the station, but cheaper: Ibis, Novotel.
In the old town, Motel One is a new build right in the centre (converted bank offices, shops on ground floor).
Hotel Basel is also a good central location, more up market.
Lots more, big and small. Depends what you are looking for.

The ultimate "price no limit": Hotel Trois Rois (aka Drei Könige / Three Kings). Oldest and poshest hotel in Basel. Even has its own Wikipedia entry.