Can I work in Basel and live im Saint Louis?

Hello, I'm portuguese and I live at the moment in Basel, I have now a work permit B in Switzerland and I would like to know if I can live in Saint Louis (France) and continue working in basel?

And about the taxes, do I have to pay taxes in both countries?

Thanks

yes you can and there are many threads here discussing exactly that. You will pay social security in Switzerland, but income tax in France.

Yes. You will need to change your permit to a G permit.

it is easy peasy. G permit is the cross border permit.