No Hindenburg disaster - I'm aware this isn't enough alone but it is key
One of the largest influences on the decline of airships was the publicity of the Hindenburg disaster. A mixture of imagery and quick reports made confidence in airships plummet.
The disaster was the subject of spectacular newsreel coverage, photographs, and Herbert Morrison's recorded radio eyewitness reports from the landing field, which were broadcast the next day. A variety of hypotheses have been put forward for both the cause of ignition and the initial fuel for the ensuing fire. The incident shattered public confidence in the giant, passenger-carrying rigid airship and marked the abrupt end of the airship era.

There are so many things that went into this that I couldn't possibly name them all but here are two that I think are important:
The US was holding onto a vast majority of the helium reserves unwilling to export this much less flammable alternative due to concerns that it may be used in military action against them and their allies. After the Hindenburg disaster this was repealed to allow exports for non-military use.
- Whether the accident would still have happened or the death-toll been so high we cannot say, but the pictures definitely wouldn't have been so dramatic.
If the news hadn't been quite so sensationalist (or even just been unable to photograph the event) then public opinion wouldn't have dropped quite so much.
- There had been many more disasters worse than the Hindenburg but the lack of coverage had meant these largely passed under the public's radar.
My suggestion would be to have the US allow the Hindenburg to use helium instead.
But that isn't enough, so what else?
Public opinion of airships is still at a high but the development of aeroplane really took off (sorry) during the second world war and their efficiency would phase out airships eventually. Aeroplanes are much better in terms of:
- Speed - An airship can reach max velocities of ~100mph
- Handling - It takes a long time to turn an airship compared to an aeroplane.
- Payload to size ratio is much lower - You need a huge ship to carry the same bombs across.
This means you're presenting a large slow moving target to the people you're attacking.
Without changing the laws of physics we aren't going to reverse any of these. We could stop the second world war...but that seems like a pretty big change to me (and another war may well have come soon after). So lets have a look at what airships can do compared to aeroplanes.
- They run off Hydrogen and helium.
- Once they've taken off you don't need to lose your fuel, you could even use (a lot of) man-power to make the ship move once you're in the air.
- You can hover. Aeroplanes had to be moving at all times or be on the ground.
Without making massive changes in history (or physics) I don't think we can stop the development of aircraft for military use. However if we use the fuel differences and say there is a shortage of oil to create fuel for aeroplanes then I think we would see similar military development focused on airships like the motherships designed to carry aircraft close to their destination but remain a safe distance from the target. This way precious fuel could be conserved. Airships would then see a lot of attention from R&D and the cheaper Helium would be preferred by commercial airlines than the expensive oil-based fuels.
You could also put a spanner in the works for commercial aeroplanes with a similarly public disaster for the aeroplane. History is fraught with disasters...and the resulting improvements made. If these improvements hadn't been made and airships were still high in public opinion then it is likely airships would have received a lot more attention.
One of the biggest developments was in suitable landing gear, meaning the heavy passenger planes could land safely. Delaying developments in sturdy and reliable retractable landing gear would have caused more accidents and decreased public opinion drastically.
In the end public opinion is key to their usage, if people won't use aeroplanes then companies won't build them. Airships would be seen as the safest way to travel and their usage would become ingrained into infrastructure. There were even plans to use the Empire State building as an airship dock. If similar buildings were constructed then calls for airports with huge runways cutting into the countryside may be seen as an excessive change.
(There are massive repercussions to this: Helium is used in a lot of scientific research, upping demand for it would make it more expensive to pursue these areas).