What do you heat the house with currently and where is it located?

For the best outcome you should contact a local HVAC contractor in your area. Don't start a landlord operation only to create a headache for yourself.

I've been a landlord since 2016 on top of HVAC service and repair, manage my own rental. It's another job... the better it's done, the less headaches you will have.

I'm not telling you what to do. I am giving you advice of what not to do. It's your choice to follow it or not. So we're clear here.

I can only provide advice based upon 30 years of experience. I am in rentals of others all summer long & sometimes in the winter, what little of that we have in Katy, Tx.

People who chime in here (other than OP) like to think I am advertising... when in fact I get calls from all over the country, that I can not help. HVAC is an in person job. I do not play guessing games with your comfort.

I service the Katy, Texas area. (The advice I provide isn't to show you how to do it, I don't recommend that, for obvious reasons if you're able to think about it.)

Just like you shouldn't try to remove your appendix on your own, you shouldn't take on HVAC without extensive experience and proper licensing. (now let's watch and see the DIY hacks, home builders that have HVAC contractors they call on speed dial flame me. LOL)

Does the basement currently have any source of heating? If not what temperature does the basement reach on days when the outdoor temperature is -15F degrees?

The smallest mini splits are 9,000 BTU. Typically a basement will have a small heating load and close to a zero cooling load. I would think 9000 - 12000 BTU would be sufficient for a 680 sq. foot space. The right way to do this is to find a HVAC mini split contractor who will do a heating load calculation. Preferably someone who has installed mini splits in basement applications.