So in theory basic auth is enough when sent through HTTPS, right?

Yes. Don't put nuclear weapons, health data or huge sums of money behind it, but basic Auth has been doing a fine job for a lot of things for a long time, and HTTPS is a complete solution (until the next time it gets owned).

If this is the case then the user would need to handle their password and my API can keep storing just the hash.

Yep. The hard part is securely delivering the generated secret to them. And making sure that, the shorter and less random that secret is, the more often it gets replaced. For a lot of not-too-sensitive use cases, a phone call and a long random secret will do the job.

In another comment JWT was suggested, maybe this could also be a solution?

JWT is a fantastic solution, and probably the first thing you want to upgrade to if your use case needs more than Basic Auth.

I'm thinking the user can worry about generating and signing the token and we could only be storing the public key , which requires less strictness when handling it, this way we can validate the token has been signed by who we expect and the user will worry about the private key.

That makes sense. Note that many popular JWT libraries will do a lot of that for you.