Jesus did not make an explicit statement on the matter, but he did seem to take Genesis as true and historical in some sense.
Consider Jesus's words in Mark 10:6ff on divorce:
"But from the beginning of creation, ‘God made them male and female.’ ‘Therefore a man shall leave his father and mother and hold fast to his wife, and the two shall become one flesh.’ So they are no longer two but one flesh. What therefore God has joined together, let not man separate."
with a parallel is in Matthew 19:1ff:
"Have you not read that he who created them from the beginning made them male and female, and said, ‘Therefore a man shall leave his father and his mother and hold fast to his wife, and the two shall become one flesh’? So they are no longer two but one flesh. What therefore God has joined together, let not man separate."
It seems that Jesus here assumes, rather than states outright or tries to prove, Adam to be a historical person at the beginning of creation.† As far as I can tell, the historicity of Adam was the shared view of all the Jews in the first century, including Paul (see especially Romans 5:12ff; 1 Corinthians 15:21f,45ff), and I don't see any reason to impute a different view to Jesus. Indeed, this passage seems to give some evidence that he held the standard view.
The key is "from the beginning of creation, ‘God made them male and female.’" Ham reads the first bit as meaning "at the beginning" (as the NIV and other translations render it and a fair interpretation of "from" in this context), and he takes the making of them male and female as happening there "at the beginning of creation."
But one could also accept Adam as historical and at the beginning (like Ham), while arguing that Jesus and the author of Genesis both believed the methods and timeline of creation are telescoped and stylized,‡ rather than a strictly historical play-by-play record as in Ham's understanding.
Under such a view, if perhaps Adam were the culmination of God's work after 13.8 billion years of creating, starting the history of mankind proper, then Jesus' understanding in the parallel passages above could be compatible with a non-Young Earth Creationist view.
In short, it seems there are multiple plausible interpretations of Jesus' words related to how he thought Genesis should be understood.§ Ken Ham has one plausible interpretation, but there are others, even among conservatives.
† Calvin also sees this as the natural reading: "[according to Jesus,] the order of creation ought to serve for a law.... And thus from the order of creation is proved the inviolable union of one husband with one wife." Calvin also here assumes, rather than proves, the historicity of the Genesis creation account.
‡ For instance, Genesis 1-2:3 presents the light being created before the ordinary sources of such electromagnetic radiation, namely the sun and stars. Interpreters going back at least to Augustine in the 400s have thus seen this passage as a highly stylized (but still good, true, and beautiful) account of creation, but not a play-by-play list of events. (Compare the Framework interpretation.)
This is not the only plausible interpretation of the creation of light and the sun/stars (Ham would give the play-by-play interpretation), but it is one with good footing in Christian tradition. One may have to choose between different views based on other factors, such as presuppositions about the authority of the Bible, the overall method of interpretation of it, one's understanding of nature ("God's other book"), etc.
§ Similar things could be said of the New Testament apostles in their writings.