Not too huge. This person[1] did some math and found that a BFR trip across the world should take 40% to 240% as much carbon as a regular plane flight. For short trips this would be worse, though, because a plane's range goes up almost linearly with fuel use while a rocket's goes up much faster than linearly.

Rockets that use solid boosters like the Space Shuttle, SLS, or Atlas V do a good amount of ozone damage as they go up but the ozone damage from cryogenic propellants of the sort that a BFR uses is pretty small.

[1]https://www.quora.com/Elon-Musk-suggested-that-the-SpaceX-BF...

Most people replies are focusing on carbon footprint, but it might also be worth considering noise (e.g. how does the sound of a rocket launching off the coast of California affect whales?) and any atmospheric perturbations caused by the rocket. Launches damage the ionosphere, which hasn't been an issue in the past because as far as we know the damage doesn't have a huge impact, and the ionosphere usually is back to normal within a few hours (or even less depending on trajectory and some other factors). If launches are happening every hour in the same area, that might cause some problems with things like radios.

Depends on how many. Flying airplanes is around 1-2% of our total CO2 footprint. Of course, for an individual, doing 1 intercontinental flight with a full plane is equivalent to months if not a year of driving a car to work every day, but in total, not many people fly.

Depending on the efficiency of the rocket and the reduced number of people doing this, the numbers shouldn't be that much higher.

Pretty similar to the environmental impact of jumbo jets. A 777-200LR burns around 145.5 tonnes of jet fuel to transport around 300 people 15,800 km. That's 31 grams of fuel per passenger-km, about 86% of the fuel is Carbon so that's roughly 27 grams of Carbon emissions per passenger-km. A BFR/BFS contains about 3,240 tonnes of propellant and allegedly will be able to transport around 850 passengers up to 20,000 km. The oxidizer to fuel ratio of the Raptor engine is currently unknown but it's likely it's around 3:1, meaning that 1/4 of the total propellant is Methane while 3/4 is liquid Oxygen, giving 810 of Methane per trip, give or take. Methane is 75% Carbon so a BFR flight should produce roughly 608 tonnes of Carbon emissions, or 36 grams per passenger-km (30% more than a conventional flight, seemingly).

However, it's possible to generate Methane fairly easily using solar power, so such launches have the potential to be nominally carbon neutral with some investment.

The BFR runs on a mix of liquid oxygen and methane, according to Wikipedia.

Ignoring obvious scalability problems, wouldn’t it be possible to manufacture that with a pretty decent environmental footprint from various carbon neutral bio sources?

It will eventually be very low, much lower than planes. The same technology that is being developed to make propellant on Mars can be used on Earth to extract CO2 from the air and/or sea water. A few squared km of solar panels on floating structures can produce enough propellant for one BFR trip a day.

The same propellant (methane) can also be used on planes, eventually, so BFR will bring many technologies advances independently of the outcome of the project.