First, since there is no condition involved, "will" is a better fit than "would". For future projections, "would" requires some sense of uncertainty or conditional dependence. @Andrew's answer covers this in detail. For example:
If the city were to expand its transit network, they would bring in an extra five million in annual revenue.
Second, you have two very different time spans here:
Over the fifty-year period: This is a duration. You would use this to express something happening over this entire time, not a single event that happens at the end of it.
By 2067: This is a time point. You would use this to point to an event, and not to a progression.
Over the fifty-year period, the ozone layer will continue to deplete, and, by 2067, carbon-dioxide concentration will be 1.5 times the current level.
Third, "predominate" is defined as "to be the largest in number or importance" by Cambridge. So, it is not an action, but rather a state. A continuous form sounds unnatural. It's like saying "problems will be existing". "Exist", like "predominate", already implies an ongoing state. The continuous form is ambiguous at best. Are you saying that they will be in the process of becoming predominant?
Putting all of this together, I would use the following:
By 2067, people aged 15 to 59 will predominate population demographics.
Now, I don't know the full context, but if you had even a vague conditional built in, you could use "would".
Many statistical models forecast a rapid growth in working population over the next fifty years. If these prove accurate, by 2067, people aged 15 to 59 would outnumber all other demographic groups.
The fact that we have a condition on the model's accuracy, the "would" fits more naturally.