Regular, every day rockets are used to blast material and parts and stuff from different countries into space to assemble a space platform in very low Earth orbit (much lower than the ISS). Because it is in such a low orbit, it will lose altitude very quickly. That's where VASIMR thrusters come in. They should already be equipped on the ISS by the time this project begins. VASIMR thrusters will reduce the need to keep blasting fuel into space so the platform can stay in orbit. As long as the platform is close enough to the Earth, it should be able to siphon hydrogen atoms off of the upper layer of the Earth's atmosphere and use it in the VASIMR thrusters.
To get power to the platform, the ISS will be put to good use. Power collected by the ISS will be beamed to the platform via microwaves. So basically, now there is a reason to keep that hunk of junk up there besides for research.
Attached to the platform will be an every day, run-of-the-mill, conventional, made-with-today's-technology crane. This should be made (not just assembled) in the USA for quality control reasons. Hooked to the crane will be a box with a door.
Hot-helium blimps or zeppelins carry cargo from the ground to way way way up in the atmosphere to within the crane's reach. Then, the blimps or zeppelins (depending on how much cargo is being carried) unload their cargo into the box. The crane then hauls the box back up out of the Earth's atmosphere and into space. Then, the cargo is used for whatever purpose. Assemble something right there on the platform, or sling it to the moon with a mass driver, or whatever.
So, except for the hot-helium blimp part, this could be done with today's technology. But it shouldn't be too hard to figure out how to heat the helium inside a blimp without the helium escaping. This is an example of better launches.
Nice idea but it will not work for two reasons:
1) Anything that moves in a low orbit around the Earth (like the proposed space platform) must move with a speed of 8 kilometers a second (i.e. about 18000 miles per hour) to prevent it from falling down, To pick up cargo from a floating platform in Earth's atmosphere (like the proposed hot-helium blimp) that platform would have to travel just as fast. If you had a vehicle which could move that fast, it would make more sense to use its speed to get into Earth orbit on its own accord.
2) Even if the crane manages to snatch some cargo from a far slower blimp, lifting that cargo up to the space platform would require speeding up the cargo to the required 8 km/s. That requires a lot of kinetic energy which must come from the crane and the space platform it is standing upon. So while the cargo is sped up, the crane and platform slow down until they fall back to Earth.