I have a new house with high humidity in the basement. Will running a dehumidifier constantly cause the new concrete to cure poorly?
1) The issues around curing concrete too quickly are where the concrete is thin and in direct sunlight (i.e. a sidewalk in the sun). The concrete in your basement has cured enough to be structurally sound. The rule of thumb I have heard is that concrete cures 85% in the first 7 days. 10% over the next 3 weeks and the remaining 5% over the next 20 years.
2) Even if the concrete had just been poured, you would not be able to make it dry too quickly with a home dehumidifier. Home humidifiers can only draw 30-100 pints (3.75 - 12.5) gallons a day and that would require you had it plumbed into a drain so you never had to stop it to empty it. Plus the moisture from the concrete just can't transfer to the air that quickly.
If you have moisture in your basement that is resulting in high humidity the risk of mold should be your concern. Run a dehumidifier and measure moisture in each basement room. Move the dehumidifier around as needed.
Nope, I wouldn't worry about it. The concrete should be more or less at full strength within 30 days of being laid, but it'll continue to cure over about a year or so.
Not only it will not cause damage, but also it helps the concrete to be more solid in less time: I used the same technique adjusting a very wet basement. The results were stunning (also the amount of water drained that I had to drop was so!).
How new is this house? The concrete should cure within a few days. In fact, you probably wouldn't live in the house if the foundation wasn't cured properly.