1 high credit, qualified tenant; 1 bad credit, unqualified tenant

[Brent Tarnow]

New landlord in Oakland, CA looking for some advice. 

We are trying to rent a unit and we have interest from a couple in which the woman has a 790 credit score, makes good money and by all accounts would be an ideal tenant.  The man has a much lower credit score with a couple of very old (but small) collections on his report.  He will also not have a job in the short-term.  

Here's my question:  Is it better to just have the tenant with good credit and finances sign the lease, or is it still better to have both of them on it?  Does it make any difference if, say, she were to leave and he decided not to vacate?  

I'm inclined to have her just sign the lease, but if he holds over does it make it harder for us, as the landlords, to assert our rights against a squatter with no lease whatsoever?  

Thanks