For when I find this thread again when googling for swiss hotels in a year or so, the youth hostel in Basel is bike friendly, they have a dedicated locked room in the basement with key access on request from reception.
I've spent the last 5 nights in hostels/hotels in 5 different countries, only one let me take the bike I to the room (cap Europe, Strasbourg). One didn't really have any bike parking options so I locked it in the underground car parking garage. One had an outdoor parking area, but as it was literally freezing and I arrived soaked, let me park the bike in their office. The rest all had a dedicated room (1 outside, the rest inside) where I could store the bike.
Quite a mix
J