I used to ride that way a lot and also struggled to find a good route. I ended up avoiding Broadway by taking a left on NE 7th (although any street from 16th to 7th should do the trick) to Lloyd Boulevard, which has a great bike lane. Once you get through the MLK light, cross the street (nice turn lane provided) and ride a short stretch of sidewalk to go under the bridge, and voila! you're at the esplanade. This always worked like a charm for me.