Would an online masters degree help me to become a professor? This question seems to indicate the answer is no but I don't know enough to see whether being accepted into a PhD program equals getting a teaching job. I attended a private four-year school at which the professors did minimal (if any) research and spent a lot of time personally assisting students outside of class. Because I've never attended a large, state school, I don't know if this is what it's like but I've heard it isn't.

I am much more interested in what I described (teaching and helping students) than in having large, impersonal classes and conducting research. 

Would an online masters degree increase my chances of getting a job teaching computer science (the subject of my four-year degree) in the manner described above?
No.

In the US, permanent university-level teaching positions in computer science, even at small four-year colleges, require a Ph.D.  As I explained in my earlier answer, an online master's degree will not help you get into a PhD program.

For some non-tenure-track positions, a master's degree with a thesis may be enough, but a terminal/professional/course-based master's degree won't be; whether that terminal master's program is online or on-campus is essentially irrelevant.

In short, if you want a permanent teaching position, you need formal research experience.