What’s the best type of durable caulk to use for stairs?
Use the picture below as a guide:

You can caulk above and in the middle of the trim pieces on the side - in yellow.
the appropriate caulk here is the DAP Alex Plus.   Yes going there because I use this caulk because it settles with the smallest bead possible and it is a nice white.   I am sure there are others just as good, this is what we use.
You make the yellow beads as small as possible.   When I caulk trim like this you would have to get on your hands and knees inches away and inspect to see that caulk was used.
The red areas - DANGER - do not caulk these
If you had to take one picture - a snapshot - yes if you caulked them the picture might look slightly better.   But with people bouncing on the stairs, people with dirty feet and shoes... Caulking here will be an eyesore in a month or two.
Caulking stairs in the redzone is literally one of the biggest tells of a &quot;bad flipper&quot;
your stairs look great.   When you do something that turns out well, leave it alone.

Side Note: Those stairs can last a long time but laminate covering will wear really quick on stair nose if people are wearing shoes.   Also laminate is really bad taking metal rubbing against it.   So if moving a shelf put a blanket or rug on stairs for protection.