Eh, it honestly depends on what the climate protestors are doing. If they're chaining themselves to people's cars, painting aeroplanes with toxic paint (so the company has to use toxic chemicals to remove it), or glue themselves to the autobahn, then they're honestly just making a mockery out of climate activism.
Moira who can't get to work because some cunt chained themselves to her wheel isn't going to become an ally when she gets scolded by her shitty boss for being late due to something she can't control. She needs to pay her landlords mortgage and put food on the table for her kids. Her old car might not be the most environmentally friendly but it might be the only option she has.
Picket parliament. Organise big costly strikes. Build gillotines and kidnap the CEOs and investors of big oil. Or you know, do something productive, like show the impact a small pond and some wildflowers can make in a green desert, or maybe work to restore wetlands.
Most people do care, but don't have the time or capacity to do shit about it. If you have the time and energy to fuck over Duncan on his way to his shitty second job so he can pay for his mortgage and his student loans, you have time to fuck over the people that have the power to help you actually make a change.