The key, in my mind, no matter what the specific solution, is that it's reusable and capable of a high launch rate.
There is no inherent reason for, say, a space elevator to be cheap and for rockets to be expensive. There are essentially four types of costs to any launch system: Development costs, manufacturing costs, per-launch costs and 'standing army' costs. For a reusable system you can add maintenance. Development costs must be amortized by by doing a number of launches, manufacturing costs must be amortized by launching the same vehicle a number of times (or using the same structure, in the case of a space elevator or fountain, etc.). Standing army costs represent the people and equipment you need just to keep up the ability to launch. It might cost hundreds of millions or more per year just to keep up your launch capability, independent of whether you actually launch anything. This, again, means you need a bunch of launches or each one will be very costly.
If you launch often enough and reuse your vehicle enough, then hopefully everything but the per-launch costs (including fuel, mission control, etc., all relatively cheap) will be minimized.
If you have a space elevator that cost $20 billion to develop and build and costs a billion each year to maintain, and can only launch, say, five payloads per year, it'll be expensive for much of the same reasons our current rockets are expensive. Conversely, if it costs the same but lifts a payload every day it'll be cheap.
So to answer you question, I think you'd be justified to pick any system that has matured and sustains a high launch rate. Take reusable rockets, a space fountain, whatever you like. One or more of them will have become mundane and commonplace in your setting, getting you the cheap launch you need.