I think Klaus just needed some time to come to terms with the fact that he was going to have a child - it’s a big thing in his life and for Klaus it doesn’t just change overnight he needs time. Klaus and Elijah are very different people and Elijah is always making these grand gestures and speeches about how he’ll protect the damsel in distress (we saw it with his devotion to protecting Elena back in Season 2 & 3, we saw it when he was hooking up with Katherine and was trying to broker peace with Klaus on her behalf in Season 4, and now we’re seeing it with Hayley as well to an even larger extent). That’s just not Klaus though - when Hayley’s life has been in danger he has been concerned, worried and wanted to help her.
I actually think Elijah tries to control Hayley (and treats her like an object) more than Klaus does. I recall even Rebekah at one point tried to tell Elijah that he needed to stop trying to dictate Hayley’s life for her and needed to allow her to make her own decisions. Hayley doesn’t need (or want) a babysitter. Hayley’s entire character arc since we met her on TVD was finding any family she had out there so it made sense she’d want to hang out with the werewolves and get to know them. Klaus seemed fine with her decision to spend time with them and gave her some space, whereas Elijah starting stalking her (which I have never found romantic - I didn’t find it romantic on TVD when Stefan stalked Elena for 3 months before they “met” either but rather creepy) out in the bayou and spying on her. It wasn’t until the explosion out in the bayou in the last episode that Klaus decided that he didn’t think it was really safe for Hayley to be out there anymore because he believed that she was probably the intended target (as he told her in this episode). Elijah has been trying to convince Hayley to move back in ever since he left because he wants to keep constant watch over her.
Klaus initially told her to stay put at the beginning of the season because he knew werewolves were banned in the French Quarter and he didn’t want to see her get herself killed. As soon as Hayley did wander into the French Quarter she was attacked by vampires and had Rebekah not shown up and killed them Hayley would probably be dead. When he heard from Rebekah that the witches attacked Hayley out in the bayou a few episodes later Klaus showed up, was concerned for her well-being, annoyed that as soon as Rebekah mentioned Elijah (someone she knew for all of a few hours) she perked up and then he carried her home. Klaus also dealt directly with Tyler (after Tyler kidnapped her and tried to kill her - which from Tyler’s POV was fair since Hayley did betray him for Professor Shane and lead 12 hybrids to their deaths) while Elijah was basically just holding Hayley’s hand and doing nothing to protect her from the guy who was trying to kill her. Once Tyler got the vampires all riled up and convinced them that Klaus was going to turn the werewolves into a hybrid army that would kill them all Klaus did try to protect Hayley (although he went about it the wrong way, I think he really did want to keep her safe) by assuring the vampires that he had given up on the hybrids and wasn’t planning on making any more and wouldn’t have a problem with them going on a werewolf hunt.
I think Klaus was just panicking. Elijah said that Hayley’s heart and stopped but the baby’s hadn’t. I think that since Hayley was more or less dead Klaus was afraid that the baby was going to die along with her so his first reaction was to try and save the life of the child (he also did try to save Hayley by feeding her some blood while Elijah instead chose to get angry and break a chair). Klaus was also more than willing to let Genevieve help (he mentioned that she did work as a nurse in 1919 so she does have a medical background and they know that she’s a witch) . Elijah carrying Hayley off to a hospital and not allowing Genevieve to try and save her (as he actually told Genevieve that she wasn’t allowed to even touch Hayley) would have probably lead to the deaths of both her and the baby. Klaus’ decision to allow Genevieve to do what she could is actually what saved Hayley’s life - not Elijah.
Elijah & Hayley have no foundation of any sort of relationship. As soon as Sophie told Elijah back in the pilot episode that Klaus had knocked her up he started making moon-eyes at her and swearing to protect her. Hayley was a complete stranger to him - and all he knew about her was that Klaus had slept with her and knocked her up so it’s always felt to me like that’s the reason he was attracted to her - solely because Klaus got her pregnant. If Elijah had met Hayley back in 4x03 (TVD) when Klaus met her I don’t believe for a second that Elijah would have been interested in her, I think the only reason he’s into her is because of her connection to Klaus. I’ve also always wondered if Elijah is aware of Hayley’s role in the deaths of the 12 hybrids (because Klaus didn’t mention that as the reason Tyler would have taken Hayley for revenge) or that Klaus saved her life from Katherine’s minion (ironically Elijah was sleeping with Katherine at the time so his girlfriend did try and have Hayley killed).
-comment on an TO article from someone who doesn’t even ship klayley, it was a response to some who said that klaus hasn’t shown any regard for hayley all season.