I wasn't sure how to proceed with the structure to hold the tread so I just built it like decking. I made boxes out of 2 x 4 and secured them level with the top of the risers using 3" screws.
It was a bit of a pain to position the boxes (which were heavy) and try to screw them in and keep it level side to side and front to back. Clamping them to the risers and drawing guide lines helped. Once that was done, I could at least climb the stairs, being careful. The cats thought it was great, they could get underneath the stairs, which at some point harbored critters (like any basement in a cold climate), and that was worthy entertainment.
Since the basement floor is 'oak' laminate, I used oak stair treads. HD and Lowes both have them for about the same price. I took a sample of the laminate and matched the stain color so I could stain the boards to match. Before installing anything though, I painted the side trim on the steps so I wouldn't have to do it after the steps were installed. This saved time in the end. I also stained and sealed the treads before installing them, letting them dry for a good four days. Using a nail gun with 2" finish nails, the treads went in fast (I numbered them on the back when I cut them since they had to be fitted), filled the nail holes with putty and eventually put a final thin coat of urethane over the existing 3 coats. They are solid. The dogs will even use the steps now, before they were too freaked out by the shakiness of them.
The A/C guy, making small talk, says "I see you finished the basement,
it looks nice. The steps are much better". I laughed since ANYTHING would be better than what was there. He said he had just rebuilt the basement steps in his house too so we compared notes on what we each did. After explaining how I did ours, he said "wow, that's kinda over-kill, don't cha think?" Hmm. Maybe.