Pictures emerged on Friday of the Arsenal midfielder puffing on a cigarette outside a nightclub.
Wilshere responded by insisting he does not have a smoking habit, while a representative of the 21-year-old claimed he had been the victim of a prank.
Still, that did not stop Gunners boss Arsene Wenger from publicly reprimanding the player.
The Frenchman warned Wilshere was putting his health in danger and he also pointed out that the England international had damaged his reputation.
Wenger said he would "have a chat" with Wilshere about the episode, but Hodgson will not follow suit when the midfielder joins up with the England squad at St George's Park on Monday.
"I trust Arsene to deal with that situation," the Three Lions boss said.
"He will make it clear, I'm sure, to Jack what his responsibilities are.
"I've had no problems of that nature with Jack, so I don't need to involve myself in it.
"Jack's an Arsenal player who also plays for England. If Arsene thinks it's also good for me to have a word with Jack I'll be more than happy to do so.
"But, as far as I'm concerned, it's Arsenal's responsibility."
With the most important period of his managerial career just around the corner, Hodgson needs Wilshere to be firing on all cylinders.
The midfielder may have only 10 caps to his name, but he is one of the few young England players at Hodgson's disposal who can turn a game on its head with a bit of magic.
Wilshere slotted perfectly into a three-man midfield alongside Frank Lampard and Steven Gerrard against Moldova, but he was a passenger in the 0-0 draw with Ukraine, where he was substituted after an hour.
Wenger dropped Wilshere to the bench for Tuesday's Champions League win over Napoli, but Hodgson has no qualms about starting with the youngster against Montenegro or Poland in the upcoming crucial final two World Cup qualifiers.
"I don't really have a concern (about Wilshere's form)," Hodgson added.
"My concern over Jack has always been whether he is actually physically fit to play or whether he's suffering from an injury.
"I've been led to believe that he doesn't suffer from that injury.
"He's in an Arsenal team at the moment with competition for places in midfield that's very strong, anyway. Will be even stronger when Santi Cazorla get fit.
"I'm hoping that he'll turn up on Monday fully fit to play, and then it'll be up to me to decide whether I want to put him in the first team and the starting line-up, just as I guess Arsene, every week, has to decide 'do I put him in the starting line-up or do I play someone else in front of him?'
"He's an important cog in our midfield thinking."
With such talent in his locker, many expected Wilshere to have had nearer 30 caps by now, but injury has restricted the midfielder since he made his England bow in August 2010.
Hodgson thinks it is important that fans remember the fitness problems Wilshere has had over the last few years when they judge the player's performances.
"He missed a lot of football, didn't he? It was a year and a half," the former Fulham and Liverpool manager said.
"Then he had to have another top-up to the operation this summer. For a young man, he's had an awful lot of serious injury problems.
"The way that Arsene has managed him since will help us."