I suggest you deal with this in your legal documentation (service agreement/contract). Personally I would bill up-front as that's better for your cashflow and most services out there expect you to pay before you use.
I wouldn't worry about customers feeling bad that they have to pay for part months for users that drop off mid-term (unless of course your per-user fees are huge) - just put it clearly in your legal doc and in the sign-up page that you are paying per user per month up-front and that there are no refunds for any unused part-months. I am assuming you are letting your users cancel at any time - if so, I would say this kind of payment structure is pretty common for SaaS businesses.
If you can avoid operating a credit system, I would - there's nothing worse than not being able to account for your income because you may have to give some of it back.
All this is said in the context that you are providing your customers with a valuable service that is well-run and they get good support - if these things are in place, in my experience, people are happy to pay fairly and don't usually get involved in penny-pinching.