There are two passages in the New Testament that, taken a certain way, would seem to indicate that he did. One is in Ephesians 4:9 where it says that Christ descended into the lower parts of the earth. This probably means that he descended to the earth, which is the lower parts. The "of" there doesn't mean that he is going under the earth. So I don't think that text warrants the interpretation that he descended into hell.
The other text is 1 Peter 3:18-20 where it says that Christ went to speak to the spirits who are now in bondage. That is, they have died—having lived in the days of Noah—and they are now in bondage; and Christ went to speak to them. Some take that to mean that between Good Friday and Easter Christ went to hell and he preached the gospel there. But I don't think that is the meaning of this text either. I think it means that when these people were alive in the days of Noah, in the Spirit Christ spoke to them through the preaching of Noah; and now they are in prison.
So my conclusion is that there is no textual basis for believing that Christ descended into hell. In fact, he said to the thief on the cross, "Today you will be with me in paradise." That's the only clue we have as to what Jesus was doing between death and resurrection. He said, "Today—this Friday afternoon, after we're both dead—you and I will be in paradise together." I don't think the thief went to hell and that hell is called paradise. I think he went to heaven and that Jesus was there with him.
So I don't say that phrase "he descended into hell" when I recite the Apostle's Creed. But study it yourself and see whether you think there are other foundations for it. As for me, though, I would say that the foundation for that particular sentence in the Apostle's Creed is pretty weak biblically.