I used to keep my dog in a shopping basket with a floor and safety harness built in. Had it fixed to my cargo rack, but then he grew up and no longer fits.

I’ve been thinking about using a wider one, but I’m a little worried he’d jump off. On the other hand, I saw this bike touring guy (Dave) do it with his dog (Mira) and it seemed to work ok.

Trailers are definitely more viable though, and far more affordable.