This is going to be a statistical mess, but since I'm not a statistician, I have no obligation to be precise.
There are about 100 million households in the US. In 2014 about 137 billion gallons of gasoline were consumed, and 90% of that was for passenger, SUVs and light-duty trucks. That's about 1,370 gallons per family per year. The EIA just published a report showing the average price per gallon in the US is now about $2.10 versus about $3.00 in 2014. So on average in 2014 the average family spent about $4,000 on gasoline and in 2015 about $2,900 per year.
Median household income is about $45,000 per year. The lower gas price, $1,100 per year, is equivalent to a 2.4% pay increase.
But this pay increase is actually skewed to the poor and middle class. If you use the numbers above, the average family in 2014 spent about 8.9% of its income on gasoline. I don't have precise data, but my pretty good guess is high income families spend more like 1% of their income on gasoline: lower gas consumption and higher income. So if gas prices go up or down, it's just not that large of an impact on the well off. But for the poor it is a very big deal.
It's even worse than that. The well-off, for instance, commute to NYC on subsidized rail lines. The subsidies come from general tax revenue and from gas taxes. That's right, they get paid to ride a train to work courtesy of those who drive, so when gas prices go up, they are even better off because they avoid the pain of the price hike AND people less off than them pay to keep them riding the rails.
It would be wrong to blame this regressive subsidy on one political party, it is bipartisan. However, it would be fair to say the Democrats are much less supportive of promoting policies that encourage lower oil prices and lower gas prices (Keystone Pipeline, hydraulic fracturing), and to that I can only say, not much skin off my nose, but it sure does hurt the constituency you claim is your primary concern.
I know the counter argument involves global warming/climate change/extreme weather. But if I'm that guy who pays 10% or more of my income on gasoline and I'm offered a choice of making that 5% of my income or paying the same or more so the rich guy living on the shore can keep his house dry, I know which one I would choose. Other than the global warming/climate change/extreme weather models being hopelessly wrong, I think that's the reason why the average American doesn't give two hoots in hell about carbon.