When I looked at it it’s only 3 hours roughly from either. I for example do Brussels to Amsterdam in a day that’s 2 hours.Basel is a very long hike from either Frankfurt or Paris. Mulhouse is a lot closer (and a cheaper place to stay than Basel), if you are looking for a base from which to do a short hop to Basel.
It does sound like a very long day trip and you would be heavily reliant on only a few Lyrias/ICEs per day. May I suggest Strasbourg? At least that way it would be plausible for you to manage day trips on the TGV/ICE/OUIGO to Paris, TER 200s to Basel or TGV/ICE/Coach to Frankfurt, with those routes being relatively frequent and reliable. Also, Strasbourg is well worth a visit if you haven’t been before.
You don't go to anywhere in Switzerland for the wild social life.
There's an Ibis very close to Basel Hbf which is convenient to stay in, and which is reasonably priced by Swiss standards. .
And also a small corner of the SBB station officially designated Bâle SNCF, I think.