Auf Schönburg is very popular and for good reasons. I would not want to make that steep walk from town with luggage. Most people will taxi or drive up.
Burg Stahleck won't have near the atmosphere - and may or may not be jammed up with pre-teen school kids on week-long outings with their teachers. It's unbelievable to us, but German schools, like the ones in the Rhine region, are often in session in July; in 2022, the Rhine schools don't start summer break until the final week of July. Since their outings take the place of the classroom, there's a reasonable chance that one or more classes will by spending the week there with you. Nothing against kids learning and having a good time... but IME any hostel with school group bookings is going to involve some noisy chaos.
Besides the castle-hostel in Diez (nearby Lahn River) I have no experience with other castle-hotels. I've heard some good reports on Burg Liebenstein, across the river and north in the town of Kamp-Bornhofen (almost across the river from Boppard.) If traveling by train, you can catch the castle shuttle from a spot near the Kamp-Bornhofen station (details from the Liebenstein management.)
Another interesting accommodation might be the riverfront wine barrel rooms operated by the Hotel Anker in Kamp-Bornhofen. It's near both the station and the KD boat dock.
I tend to think of castles as better places for touring than for sleeping. For a fine Rhine Castle tour, do pay a visit to never-destroyed Marksburg Castle (in Braubach, north of Kamp-Bornhofen.) There are good hotel and apartment options in Braubach as well.