At four months I had my daughter ride in a trailer with her car seat strapped in, facing backwards. She didn't wear a helmet, but the car seat offers far more head protection than a helmet could. You do have to go slow and lower the tire pressure to ease the bumps. I estimate about half the speed that I can make without trailer.
Once she was old enough to hold her head up with a helmet, she started riding in a seat on the rear rack. This is much nicer because she is close enough to talk to me. I prefer to have the seat in the back instead of front because I didn't want all of the extra weight affecting my steering.
She's now two and loves riding with me. I take her to day care on the bike just about every day. We were able to ride most of the winter except when it was too cold and icy.
I still have the trailer, but she doesn't like it, I think because she is too far away from me to interact.