Can it be viewed as a negative for PhD applications in the US if I have children? I am currently working on PhD applications in Engineering to start in Fall 2020. I am in my late 30s and have one child. I am planning to apply to at least one quite highly-ranked school in the US.

I am wondering if it might be seen as a negative for the admissions process that I have a child? I have heard that top US PhD programs are not looking for 'balance' and that they typically expect candidates to give everything to the program. However, I have heard of people doing PhDs with kids (even having kids during a PhD program) and still being able to complete without any major problems.

In particular, should I avoid mentioning that I have a kid in my application/personal statement? Or, conversely, should I mention it? Would it be seen as dishonest in some way, if I gain admission and then turn round and tell people I have a child?
I doubt that you will find any general problem in the US, though it is possible that some individual professors might take issue. Some labs, for example, expect long continuous hours at the bench. 

I don't know of laws in the US governing this, but there may be some. 

There are tons of grad students with children. I had two kids in the final years of my doctoral study. But, you don't say if you are a single parent. In such a case, I don't think you would find discrimination, but I do think you would find it hard. But then pretty much anything and everything is hard for single parents. 

In many subjects, of course, you can actually work while tending a child. Mathematics, for example, or Literature. Even CS folk can do quite a lot at home on a laptop. 

And note that there is nothing special about academia with respect to "giving your all". Many employers expect exactly that. 

My suggestion is that you don't mention it in applications, but because there is no need. 



And, of course, professors may have kids and are still able to work effectively.