There are a couple downsides to send a single thing.

The final propulsion needs is mostly the same, so you won't pay more to push stuff to Mars than you would to boost that stuff from LEO to a Mars transfer trajectory. The mass and delta-v being the same, the propulsion budget is the same.

Say, if you assemble it and then something happens and you can't slow down and enter Mars orbit, you lost the whole thing.

If the station is assembled, it's shape won't probably work well for atmospheric braking and would require a bit more fuel to slow down. A single module could, conceivably, dive deeper into the atmosphere and shed more speed with that.

Acceleration loads on the assembled thing are also to be considered. If we decide against chemical rockets, the question is mostly moot - acceleration will be low, but constant.

If we send multiple modules separately, we end up with a lot of propulsion capacity that, with some clever design, can be reused for landings and orbital ascents. You can never have too many of a single thing when you are that far from Earth.

The upside is that it's easier to assemble everything in Earth orbit and it'd be easy to teleoperate robots.

Having said that, I suppose an extra week living in the Earth-Mars transit vehicle while the arriving crew assembles their home is not a big deal for someone who already spent 3 months in transit.

edit: if we are to do that, putting a station in a cycler orbit would probably be more interesting. That way you depart Earth with supplies and rendezvous with your living quarters for the trip. The cycler can have the shielding to protect astronauts on their way.

There is no reason for the whole station to land on Mars. Approaching Mars the whole thing would split to start breaking. Big parts of it (bigger than a BFR) with payloads--half-baked assembled constructions--would separate and enter Mars atmosphere for landing. Then the station itself will be less heavy for breaking using the gravity of Mars and its orbit for that.

I think we were talking about a space station. If it ever lands, I assume it'd be unintentional.

Yep. SS. Also apart from the tech stuff, it's important to consider financial and administrative issues. Building the thing near Earth allows making it attractive for sponsor/ad money and save construction and preparation time starting earlier in space using Falcons. Meaning existing plan does not allow any payload launched unless BFR is built. Not to mention spending fewer resources to build all that BFRs required launching all payload simultaneously during the launch window.

P.S. BTW, ISS has limited service time left. So there is a need to plan construction of a new Space Station on Earth's orbit in any case.

You don't need - and you don't even should - launch all pieces at once. You can launch them into the same orbit so they meet at some point. It's easier to make the parts meet in Martian orbit because the distances allow for finer corrections over longer periods of time. It doesn't matter if some pieces arrive months or years before others as long as they are all there when it's time to assemble the station.

A Falcon Heavy can take about twice as much cargo as a shuttle could to the ISS. This means the launch costs of upgrading/refurbishing the ISS, or even replacing it, just got much lower, and the whole process requires half as many launches (and the modules can, now, be bigger than the shuttle payload bay).

I assume the BFR should be able to take, with a non-reusable second stage, almost a whole space station, ready for assembly. Or to carry large modules, such as a centrifuge, so that astronauts don't need to be continually exposed to microgravity during their entire stay in orbit.

And with the full BFR, it can return ISS modules to the ground for study or refurbishment which, I assume, is valuable as it allows the study of structures and materials exposed to space for decades, as well as reflight of existing and already validated components.

We do build them all on earth, orbit them and send them to the Moon or Mars etc and land/orbit them for their tasks.

A large structure - too large to launch as a complete station should be orbited here as many separate loads and then assembled to form the station, and then sent to Mars - as you suggest. It is not practical to send the loads to Mars and assemble there.

Every item from earth has a huge cost. Items made from loose asteroids, the Moon, Mars etc may well be a lot cheaper. You must bear in mind that elements(metals, Oxygen etc) would need to be extracted and made into raw materials suitable for whatever you want to make. That means each one would need to be screened about the relative energy economics. Few items from asteroids could be used as they are found.

The problem with this is the delta-V requirement to actually send it to Mars. Even with in-orbit refueling as proposed by SpaceX you'd probably require a big rocket and/or a lot of fuel to send something really massive.

Ion rockets might be used but that would make the trip take forever and would be very costly.

There do exist very slow low-energy transfer orbit trajectories, but AFIAK those require you to start at various Lagrange points. So maybe it could be done but you'd have to build it at a Lagrange point, destroying the cost savings, and it would take forever to deliver to Mars.

Sending simultaneously many BFRs basically with the same payload right from Earth I'd say would require even more fuel, lot's of investments to build that many BFRs and then construction need to be started from scratch after the launch on Earth and on Mars after unloading. So all the time and money will be spent to build a huge amount of rockets.

The major cost is lifting mass out of Earth’s gravity well, so if you really want to cut costs we require sources of materials which don’t come from Earth. Once you have this, and build your station, why send it to Mars? Keep it here, or send it to a planet with resources we can use (a moon of Jupiter might work).

You’d need a compelling reason to do this, since the cost from R&D to launches would be in the trillions and take decades.

IIS has not so many years left to live, BTW. Space Stations as posts on orbits are good for many practical and scientific reason.

Absolutely, but to me that’s an argument for space stations in orbit around Earth, not Mars.

Agree. Though one does not exclude another. Starting with a new ISS to test ideas and tech and extending it later to SS for Mars would be logical.