Sometimes a trash bin is located near the door, so I'll use the same paper towel I used to dry my hands to open the door, hold the door open with my foot, then throw the paper towel in the bin. But these make hygiene so much easier:
My favourite is the kind of S curve that some places have, so you just walk in, but it's private enough that people can't just leer from the hallway or whatever I'm not actually sure what we're accomplishing with doors here unless it's a very tight space I guess like if the bathroom is near the area where patrons eat at a resto? Yeah I get that, door away. Sorry for rambling.
I grab it with the edge of my shirt. While it's not ideal, my shirt will be washed later and it spares me having to deal with risk of fecal particles on my hands where they can immediately reach my face.
I just use my shirt tail or sleeve, haven't opened a door with my.bare hand since February 2020. (Yes I do realize COVID isn't spread by touching really but if it's one less risk I can take along with masking I will do it).
The correct way to handle this would be to use a disposable paper product. Most places still have a paper towel dispenser along side the air hand dryer, you're supposed to use that.
Besides the point, most people don't know how to properly wash and dry their hands. There's a technique to both that actually improves cleanliness a lot and reduces overall waste.
I never use air based hand dryers. Paper towel for life. Some places use maze patterns instead of doors, which I generally like but usually requires some extra work with air handling to make sure the bathroom air stays in the bathroom, and a bit more floor space to provide the room for the maze pattern. Those restrooms are usually the ones without paper towel, I don't mind, I just have moist hands for a few minutes afterwards.
All of this can be googled. So I won't go into more detail, but the majority of people couldn't possibly give fewer fucks about handwashing or hand drying properly. So I expect most won't even try to learn how to do things better, ever. They just go with whatever their parents taught them as a child and never question it again. Bluntly, your parents probably did the same, so you're probably working off of 50+ year old advice on hand washing.