I'm a first-time asparagus grower this year too. Mine are in trays and as with you, they're tall and falling over. I'll be putting them in my garden towards the end of the month.

if confined to a cup or smallish area, will fall over. Your backyard site looks adequate, though. Maybe not enough sun?

(why I gave up growing asparagus in my urban backyard)

and then graduated to 18" deep so that the season would extend. You may not have planted deep enough. Do not cut the asparagus for at least 3 or 4 years and then sparingly. The bed is usually established in 7 years. I didn't feed my bed last winter and the weeds and grass really took over. I had a VERY small season this year. I have to clean it and feed it well this fall so that next year will be robust. Oh, at the end of the season when the ferns are completely dead, that is when you cut the plants to the ground. Make sure that the cuts are clean. I had a friend help me last season and he used little pruners and tore the plants. I think that maybe they got diseased. Okay, that's all I know.

Figure on not cutting any until year 3, at least, and of those only the ones as thick as a Sharpie permanent marker. I was gonna say as big as your little finger, but that varies. even so, you do not cut all the ones that look good enough to cut, those roots need to grow a bit more in the following years.

You can google when to top dress them, and when to cover the bed if you live in a cold area..they will grow just fine.
My first and 2nd year plants tended to fall over, the stalks were too thin for all the frowsy leaves, but nothing bad came of it.

yours look pretty standard for young plants. The fern-like tops do get larger and larger as the plants age and will flop over. You can just let them do that, although when I was in my last house I propped them up with cheap tomato cages to keep them off the pathway so nobody would trip over them. I cut them off after frost killed them back and then mulched the bed for winter. The advice from another poster about setting your plants deep is recommended by all of the gardening books; if you didn't do this maybe wait until after they die back from frost and lift them gently to dig the trench a bit deeper.

alongside the road south out of Kettle Falls, along the eastern shore of upper Lake Roosevelt/Columbia River, was a treasure trove late spring on the southwesterly facing slope of the roads berm.
Free Range Asparagus !