Understand the difference between a fear and a phobia
I think anyone who has ever been around me when I've come into contact with my phobia knows I don't have some run-of-the-mill fear. Just saying.
Don't laugh at their phobia
You can perhaps call me 'lucky' that my phobia is one of the common ones. And by lucky I mean that people won't laugh at it. But you can have a phobia of absolutely anything - and the more obscure the more embarrassing it can get. Don't ever laugh, no matter how mad it might sound.
Have a chat with them
For some phobia sufferers it might be difficult to open up, however I've personally always been really touched by friends who make the effort to understand exactly how mine personal manifests itself. Things to ask include how severe it is, what triggers it, any avoidance strategies and how they react when they come into contact. If you want to approach the subject a good opening line "I want to be fully supportive of your phobia, and would like to chat so I know what I can do to help avoid contact and also be prepared for what to do when you possibly do come into contact".
Don't suggest that film you know has their trigger or remember to hide an ornament away if they're staying for that weekend. It's not going to effect your life too much if you have to double check your house for triggers but for them, it's the difference between a panic attack or not having a panic attack.
Don't question any avoidance strategies they take to protect themselves
I avoid situations that probably do look bizarre to a non-phobia sufferer. But most individuals who have a phobia will have a list of places or types of situations they'll avoid to protect themselves and avoid contact. Most of my avoidance strategies were developed because of a previous experience that led to contact, so I know there's a chance of contact happening again. Also, if they want you to proof read a book, magazine, recommend a film, or quickly walk round a shop to okay it for them, then do so.
(As an aside: I turned off images in my browser while researching for this post - that's one of my avoidance strategies)
For me, contact with my phobia usually results in running out of the situation and sobbing. I can also feel uncomfortable returning to the room where it happened for some time after. But other phobia-sufferers might have a different reaction - from full-scale panic attacks to fainting. If you know how they react in advance and have had a chat about what you can do if contact happens, then you'll be in a better position to offer support.
Don't tell them to 'face their fears'
For starters, a phobia is not a fear (though it is similar). For seconds, phobias usually require the help of a trained professional for them to be overcome. That's partially what makes them different from fears - they can't be faced in the same way. If your loved one does decide to go through treatment, be there for them but let the trained professional do their job. The only time you should intervene is if your loved ones phobia is so severe it needs treated and they're refusing.
(Another aside: I did try and face my phobia by forcing myself to look at images - it made it worse).
There's a lot of phobias out there and you can't possibly add trigger warnings to everything you post online. However, you can add a trigger warning for all common phobias. If you look up trigger warnings, it's recommended that they are used for common phobias however in practice I rarely see this being done (even by people who do use trigger warnings otherwise). Common phobias to be careful for include snakes, needles, vomit, spiders, sharks, and blood. Think twice about using these images without some kind of warning.