Final answer:
DNA ligase is the enzyme responsible for joining Okazaki fragments together in eukaryotic cells. A mutation in DNA ligase is likely if there is an impairment in the joining of these fragments during DNA replication.
Step-by-step explanation:
In eukaryotic DNA replication, the enzyme responsible for joining Okazaki fragments together is DNA ligase. When dealing with a cell strain that exhibits impaired joining of Okazaki fragments, it suggests a likely mutation in DNA ligase. DNA polymerase I is responsible for removing the RNA primers and replacing them with DNA, but not for joining the fragments together. The functions of DNA polymerase β, DNA polymerase δ, DNA polymerase ε, and RNase H1 are distinct from the role of DNA ligase and do not perform the function of joining Okazaki fragments. Therefore, in the context of the question, the correct enzyme most likely to be mutated if there's an issue with the joining of Okazaki fragments is DNA ligase.