What is the biggest problem the Darlings have?

The following answer is based solely on my opinion, but I would suggest the biggest problem with the Darling family is that the parents are simply not very attentive. Although both Mr. and Mrs. Darling are aware that a strange boy has been attempting to enter the nursery, they leave the kids unattended in the nursery and head out of the house for a dinner engagement at the beginning of the story. If Mrs. Darling had followed her instincts and stayed home, if Mr. Darling had allowed Nana to remain in the nursery, or if either parent had paid attention to Nana's warning barks, Peter Pan and Tinkerbell would most likely not have entered the nursery. The fact that the parents employ a dog as the children's nurse also indicates a lack of attention. Although Mr. and Mrs. Darling clearly love their children and realize some of their parenting errors during the novel, had they been more present with and attentive to the children in the first place, it's likely that Peter Pan would have had to look elsewhere for a mother figure, and Wendy, John, and Michael would have remained contentedly at home in the nursery instead of journeying to Neverland.