I don't know Basel, so can only point you to the two portal sites where dog sitters list their services: https://petsitting24.ch https://www.betreut.ch/tierbetreuung
Note that both sites require registration and payment of a membership fee to get access to carer's details. You can browse, however, to see if perhaps someone meeting your criteria is listed.
Beyond that, try Rent-a-Rentner, a portal where folks who are retired offer their services - for all manner of jobs, not only pet care: http://www.rentarentner.ch
And of course, place an ad here on EF.
EF member Peg A used to offer dog care in Basel. She hasn't been active here in quite some time, though - don't know if she is still doing so.
Do be aware, however, that anyone caring for a dog in return for pay is subject to training course requirements. What courses need to be done depends on the number of dogs cared for at one time. With 5 or fewer, the carer only needs the SKN courses, the same courses that an owner is required to take. Makes sense that anyone being paid to look after a dog have the same knowledge as an owner is required to have.
Caring for more dogs requires more in depth courses, the FBA or Tierpfleger diplomas.
You can read more here: Requirements For Dog Care Providers
The reason I am bringing this up is that the onus falls on you, the dog owner, to ensure that the person you pay to care for your dog is in compliance. You are the one who can get fined, your insurance might refuse cover in case of an accident if it is found that your carer is not compliant.
Unfortunately, many dog carers are not compliant, even those listed on public sites. Again, the onus on you is to check for credentials as the liability rests with you.
The cost of in-home care is all over the ball park, by the way.
To give you an ideal, the average for someone in my area who takes a dog into his/her own home is 100 per day. The average for a carer who stays in your home to care for your dog is 200 per day. I have no idea what costs are in Basel.
I hope you find appropriate care. I know how difficult it is to find someone willing and able to offer the kind of care you need.