@KarlSchwede thanks for replying. If I understand correctly yes. This is proposition 3.1 in arxiv.org/pdf/math/0307094v2.pdf . If I knew normality, I would have that the quotient is still normal, because the group acting is finite. But for seminormal I don't know how to prove that also the quotient is seminormal.

@Karl Schwede thank you for the references but still I need the symmetric product case not simply the product. For smooth curves the symmetric product is still smooth and I would like to say the same in the seminormal case assuming that I know that the product is seminormal.