Off topic, and hopefully you already know this, but just in case:
Bull Terriers are on the restricted list as potentially dangerous dogs in both Basels. This means you have several hoops to jump through to gain the special permit necessary to be able to keep your dog in BS or BL.
The regs are here for Basel Stadt: http://www.gesundheitsschutz.bs.ch/t...che-hunde.html
And here for Basel Land: https://www.baselland.ch/hunde_gefae....280962.0.html
And now back on topic:
I would contact Tierschutz Beider Basel to try to book care for your dog. They offer dog care as well as serve as a shelter for homeless dogs. These folks have a good reputation - and are well aware of the issues surrounding listed dogs. http://www.tbb.ch
I also know of one pet sitter working in the Basel area, but I don't know if she takes on listed dogs, or not. Will PM you with the info.
ETA with my usual sermon-type comment
When one has a listed dog one must be sure to follow the law at all times. (As should owners of dogs of any breed - but the reality is that owners of listed breeds tend to have a target painted on their backs and so need to be doubly careful...) So please make sure that anyone in Switzerland caring for your dog fulfills the legal requirements for a dog sitter. Requirements For Dog Care Providers