"He seems to be rich" means he appears to be rich. "He seems not to be rich" is incorrect but means "He does not seem to be rich" when written correctly, but the problem is that one sentence is clearly negative and one is not. I can't say that the sentence "He seems to be rich." is actually correct for whatever purpose you are intending based on just those two sentences alone.

He seems to be rich (= there is some evidence that makes me think that he is rich ... but there is some doubt) He doesn't seem to be rich (= there is not evidence that makes me think that he is rich ... but there is some doubt) He seems to not be rich (correct but slightly unusual = there is evidence that makes me think that he is not rich)

He seems to be rich is correct.

I honestly don't know what the other people answering meant, but both your sentences are perfectly correct, though they mean exactly the opposite of each other.

He doesn't seem to be rich. The other one is correct