At the very end of the movie, after spending the whole thing warming up to Dawson, Basil doesn't know how to ask him to stay, until a new client comes in. What follows is a surprisingly heartfelt, yet roundabout way of Basil asking him to stay.
Basil: Allow me to introduce my associate, Dr. Dawson, with whom I do all my cases. Isn't that right, Doctor?
Dawson's first meeting with Olivia. Upon arriving in London, Dawson finds the poor girl lost and crying in the rain. He comforts her as much as he can and offers his help, even going out of his way to escort her to Basil's office and sticking around to make sure she's all right. This whole exchange makes it clear right from the start what a good man Dawson is, bumbling aside.
Dawson: Hmm... I'm afraid I don't know any Basil.
Olivia:[gives him puppy eyes]
Dawson:[smiles] But I do know where Baker Street is. Come along, we'll find this Basil chap together.
When Basil, Dawson and Olivia go to the toyshop to try and track down Fidget, Basil is standing on the ledge and Olivia can't make the jump from Toby, so he reluctantly holds his arms out. When she jumps to him, he briefly holds her close before suddenly realising what he's doing and quickly puts her back down, and Olivia happily skips off.
Also, when Basil notices how upset Dawson is, he immediately stops ranting and realizes that maybe he Was Too Hard on Him. While he doesn't actually apologize for his rant, he does manage to comfort Dawson. It's quite Adorkable.
Basil: Dawson? I say, Dawson, old chap?
Dawson: Poor girl. I — I should have watched her more closely.
Basil: There now. It's not entirely hopeless. (Beat) We'll get her back.
For the entire movie, Basil can't get Olivia's last name right because he's too focused on the case (despite getting it right when talking to her father). When the case is solved at the end, he gets it wrong again but by the warm look in his eyes and the way Olivia smiles and shakes her head, it's become something of an inside joke for them.
Flaversham and Olivia's reunion about halfway through the movie...at least until Ratigan ruins it a moment later.
Basil's little smile when he's pedalling up to join the others after his Disney Death.