There is a very good tool for role playing games, called the "Same Page" tool. It's used when a gaming group comes together, to ensure that the people that are going to be spending significant amounts of time on a game are looking for the same things (or at least similar) out of the game.
If you didn't do something like this at the start, then you will need to level set with your students, especially the ones that are behind. Nobody expects you to work for free, but if you make it easy for them to avoid paying, many will (Either out of forgetfulness or neglect). You would need to emphasize the necessity of on time payments on a per session (or other arrangement, you could do something like $20 per session or $16 per session for a 10 lesson payment, things like that).
You may have to get to the point of withholding services. You are a commodity (knowledge of a subject) that they are attempting to utilize to better understand. Your time = money (Trite, but very true). If you spend your time and knowledge and do not get anything in return, then you can't pay bills, eat, etc. Also, you say that many don't pay until the end of the course? What happens if they just stop responding? After all, they aren't being tutored any more, what is the incentive to make up the back fees?
If they can't pay it all, offer to work for them. If you charge $20 (I am assuming US dollars) per session, let them know that you will work with them, they can pay $20 for the session and $10 makeup fee until the back fees are paid, and that in the future all money is due at the start of a session.
If they ask why, you can respond:
Unfortunately, I have had some students that have not paid for the tutoring service, which places a burden on me. To prevent this and enable me to continue tutoring, I need to implement this system. I appreciate your understanding.
Or something along those lines, whatever you are comfortable with.
In the future, I would have (if you don't already), have a written contract that spells out the responsibilities (Of both you and the student). You agree to show up on time, be available, give notice of obstacles, etc. They agree to show up, do the work, pay, etc.
The downside is that you may have to eat some fees or lose a student or two. If the payment structure was unclear, and now you are enforcing it, some may feel that it is a surprise and not like it. But bottom line, you need to do what you need to in order to keep tutoring and still make the money you need.