Even ignoring energy used in transporting canned/bottled beer to stores, you can save significant amounts of energy.
Since a growler is reusable many times, I'll ignore the energy cost of making it (if making a growler consumes 10 times more energy than to make a bottle and you use it 20 times, it's roughly equivalent in energy to making one half of a bottle)
According to this article at Wired, making a "virgin" aluminum can consumes around 2.1KWh of electricity, but making a new can from recycled aluminum consumes 96% less energy, 0.084KWh. Making a new glass bottle consumes around 1.1KWh of energy, while a recycled glass bottle consumes 26% less energy, or .81KWh.
45% of cans are recycled, so the average energy to create a can is:
0.084KWh * .45 + 2.1 KWh * .55 = 1.19KWh
25% of bottles are recycled, so the average energy to create a bottle is:
.81KWh * .25 + 1.1Kwh * .75 = 1.03 KWh
They work out to about the same, let's split the difference and call it 1.1KWh
Since your household goes through 1 six pack a week, that's 312 bottles/year, or around 343KWh of energy went into making those bottles (the energy consumption would be a bit lower if you're diligent about recycling -- though much higher if you don't recycle cans)
I couldn't find average household energy consumption numbers for NYC, but the average energy consumption for NY state is 603KWh/month.
So, in conclusion, by consuming beer from growlers instead of bottles or cans, you're saving around 17 days worth of your household electrical consumption (or around 4.7% of your annual consumption). (If you live in NYC in multi-unit housing, your monthly energy consumption is likely to be lower than average so your percentage energy savings may be higher).
Like all recycling/reuse efforts, the effect becomes much more more significant if many people make the switch.
In 2012, cans had 52% of the market, and bottles had 36% of the market -- the rest is draft beer
(these same numbers quoted here)
The annual beer consumption in the USA is 24186 x 10^6 liters, or 8.2e11 ounces. I don't know the mix between 12oz, 16oz and other odd sizes, but based on what I see at the store, I will guess 75% 12oz, 25% 16oz and ignore the less common sizes.
Therefore, I'll estimate that each can/bottle contains an average of .25 * 16 + .75 * 12 = 13 ounces
8.2 * 10^11 / 13 = 63 billion bottles and cans.
Using the 1.1KWh per container energy costs from above, that's 69.3 * 10^9 KWh
In 2012, annual residential energy consumption was 1.3 million thousand MWh, or 1.3 * 10^12 KWh
In conclusion, if everyone stopped using bottles/cans and used growlers, that would save around 5% of the residential energy used in the USA.
That's much higher than I expected, someone should check my math (I tried to reasonably preserve significant digits, but I cut some corners here and there while cutting and pasting). Bulk pricing (1000+ bottles) for beer bottles seems to be around 50 - 60 cents/bottle excluding shipping, so I suppose it's reasonable that 15 cents of the price goes toward electricity to make the product.
Of course, there's no way that 100% of consumers would switch to growlers, so adjust the numbers accordingly (i.e. if 25% people switch,then the energy savings would account for around 1.3% of residential energy usage).
And other factors would come into play, right now around 90% of recycled glass goes into making bottles, with fewer bottles being made, some of that glass would need to be diverted into other, possibly less suitable uses. At least initially, it might be less convenient, and people might use more energy traveling farther to find beer in a growler, but eventually they'd be refillable everywhere. Perhaps spoilage would become more significant if grocery stores had to discard beer that expired before the keg was empty, and if consumers didn't always drink the entire growler before it expired.