Running a light that is obviously red is very dangerous, so it is rarely done on purpose.
Running a late red on purpose is very dangerous and nobody does it on purpose. And people who do it by accident aren't going to be any more deterred by a red light camera, because if they'd realized they it was a red light they would have stopped, with or without a camera.
However, in some cities I've been in, during rush hour, at busy intersections I've observed a pattern where the light is green and cars flow through the intersection, the light turns yellow, and cars continue to flow, the light turns red and cars keep flowing through. Its not particularly "dangerous" because traffic is heavy enough that the opposing lanes are all stopped and backed up themselves, and they won't start while the intersection still has an uninterrupted stream of cars flowing through it, even though they might have a green light.
Another related pattern is left turns on regular green lights, where by law (at least where I live), a car can 'establish' itself in the intersection during a green (or yellow), and then when oncoming traffic clears, it completes its left turn. It might enter on a green, and have to wait until yellow, or even red before completing the turn. And this is legal.
But again, I often see cases where multiple cars complete the turn, even cars not 'established' in the intersection, but several cars queued up, the last several are entering the intersection on a red, and again this is only mildly dangerous as opposing traffic has been stopped, and is waiting for the intersection to clear, and again the cars running the red are part of an uninterrupted stream.
I've seen it in some cases, where more than half the opposing traffics green light is blocked by a constant stream of red-light runners. Each one 'secure' that as long as they are in the stream, opposing traffic isn't going to start.
Red light cameras effectively curb this undesirable behaviour.
This is supported by a large amount of data that show that accident rates either stayed flat or increased in almost every case
Provided they don't mess with the timings, there may be a rash of relatively miner rear ender as drivers adjust to the idea that they can't run red lights anymore. And this isn't necessarily a 'bad thing'. A bit of mild short term pain for long term gain, and a reason why looking at the accident rate doesn't tell the whole story.
Here's the real question - why do people continue to push red light cameras for safety when there is real data that shows that red light cameras have no net positive effect on safety?
Red light cameras as revenue generation is asinine. And red light cameras for 'safety' is dubious at best.
But they can improve traffic flow by enforcing the timings as displayed by the lights. (per the scenarios above). And indeed they are an appropriate solution here.
And I generally support responsibly installed red light cameras. (Ie those installed without tampering with yellow duration).
The average responsible driver will never run afoul of them.
I despise speed cameras though. (And not because I get speeding tickets, but because they are misused in ways that are just disgusting... I was recently in Melbourne, and the tolerance there is crazy low. They automatically ticket people for doing 62 in a 60. And they'll do things like set up traps just in front of the 100km/h sign -- and ticket people transitioning from 50/60km/h to 100km/h as they approach the sign (since its not technically 100km/h until after the sign...)
That's not about safety. That's not about traffic flow. That's just revenue generation.