I started to post a link to Bushtec trailers but their website is down right now. If you have a touring bike, you can probably find a hitch from them and get a bike trailer. They make one just for bringing your four-legged friend along on your rides. It has a rear loading ramp and the lid opens to allow the dog in. When you lower the lid, it has a glass canopy (for lack of a better word) so the dog can stick his head up and see what is going on. The canopy also is open at the top so there is always a fresh air supply. I know dogs like their head in the wind, but the canopy's windshield effect helps protect their eyes from flying debris and insects.
And the fact that a trailer maker saw a big enough market for this tells me that bikers generally like dogs pretty much too, much like those of us in this forum.
Right now, I have no pets because I live in an apartment. I like larger dogs (prefer Alaskan Malemutes) and I think it would be wrong to keep them cooped up in an apartment. They need room to roam, even if it is just a backyard occasionally.