Interesting to review this in light of what actually happened. SpaceX didn't call it the BFR, but renamed it the Starship/Heavy. The cost of development of the rocket itself came in at about the number mentioned below (if memory serves of one estimate I saw).
A moonbase will likely be constructed, as a return to the moon is now a NASA priority. Elon Musk wants to develop the Moon too.
I would favor a new ship be constructed that would launch from the moon to an EML collection point. It would be huge and capable of transporting thousands of colonists in relative comfort and safety. It would likely be a torus that could be rotated, and would have ample shielding from the deadly cosmic rays in interplanetary space.
The lunar base could supply some of the materials for the ship. The rest could come from the Earth. The "Battleship Galactica" would launch from an EML spot, which would require a minimum of Delta V to get into a Martian Capture orbit. From there, the crew could disembark via the Starship on multiple visits to the Martian surface. It could refuel there as many times as needed in order to bring all the colonists to the surface of Mars.
One hundred thousand colonists could traverse the distance from the EML location to Mars on each Holman cycle of about two years. Thus it would take about 20 years to get a million colonists on Mars.
There was an article on NextBigFuture, mentioned courtesy of Free Republic, which was about the BFR's development cost. It is estimated to be anywhere from $2 billion to $10 billion.
Given the great potential of this rocket, it would be of the utmost national importance for the government to grant enough business to SpaceX so that they can have the necessary funds to develop the rocket.
You could do this with a moonbase project. It was estimated by NASA to be a project that was doable under the current NASA funding scheme. The actual numbers ran out to about half of the Apollo project.
If a moonbase was turned into a commitment, such as with Apollo, it could be done in ten years at a price that should be even less than the NASA number. Of course, SpaceX would have to win a competitive bidding war. But it should be done as a national priority.
The government is not likely to do this however, as funding tends to get spread out over a number of states. This doesn't work well with one company getting the lion's share of the business.
What purpose would a moonbase serve? It could process lunar regolith into fuel. The fuel could be used for deep space missions.
The most likely fuel from the moon would be oxygen. Oxygen is not a fuel, but is a reaction mass. For all intents and purposes, it is the same thing.
For the raptor engine, which uses methane, oxygen would be close to 80% of the reaction mass. In such a scenario, it would be profitable for the BFR to land on the moonbase, load the oxygen, and transport it to a refueling depot at the L1 Lagrange point.
The advantage of using these points is that it takes less energy to get to the Lagrange point than to the moon itself. Morever, the big rocket could be mostly fueled up for a trip outbound to Mars and other destinations. One big rocket can service the Lagrange point, and provide extra for more ambitious missions.
Almost all of a rocket's launch mass is fuel and oxygen, so if you can get it elsewhere as opposed to the Earth, you can vastly improve access to the Moon and points beyond.
