What makes a MSc in Computer Science difficult for someone who has no specific background but 10 years of experience as a programmer?
It sounds like you fell into a couple of common traps at the same time.

Trap 1:

I am good at front-end web development, I am surely good at computer science as well

As you likely learned by now, it is possible (quite common, actually) to be a terrific programmer and still struggle with fundamental CS topics, such as algorithms. There may have been a time when programming was really applied algorithms, but today the skills required to write your average web application is relatively disjoint from what you learn in CS 101.

Trap 2:

I did decent in (undergrad) maths and physics courses, surely those (master) CS courses will be ok.

This is a combination of multiple fallacies I have seen. Firstly, you may have not taken into account that master-level CS courses are, well, for students on master-level. They assume a strong command of the basics, which, by the sound of it, you lack. Further, doing ok in maths and physics is a good indicator that you'll do well in maths and physics, it by no means qualifies you directly for a CS master.

Trap 3:

I know what I need to know, I just applied to the master because I need a better degree.

You say that:


  [I] figured it would be great to have a degree so as to look like a more legitimate programmer than what my anthropology B.A. degree might imply


I find this statement very concerning, as it seems you are more interested in getting a degree than in learning CS. If that is the case, you should seriously reconsider whether the degree is actually worth the trouble. Frankly, for most programming houses I am in contact with, 10 years of experience count for more than a master's degree anyway.

Now to answer your concrete question:


  If you were me would you quit the program? I'm not sure if it is a lack of a background in undergraduate computer science, the fact that I am an older student who is nearly 40, or a lack of aptitude for this subject.


I would not assume that you are just "not good enough", and the age shouldn't really be an issue either. It definitely sounds like you are lacking background, and the fact that you are doing the master part-time surely isn't helping either. All in all, that does not make for very favorable conditions.

Nobody but you can tell whether you should quit, but given the information, I would re-evaluate whether (a) a master is achievable for you, and (b) whether getting a master's degree is actually worth it for you.