Generally speaking, if you wish to be involved in designing, proposing, and interpreting experiments, you need to be a PhD. (This isn't specific to synthetic biology.)
That's not a hard and fast rule. There are certainly Masters and Bachelors out there who will propose experiments and do analysis, and any organization worth their salt will consider intelligent proposals for experiments from technicians. However, a PhD is usually taken to be the sign that someone is capable of designing and executing an independent research program. -- In some respects, that's what a PhD degree is. It's a certification by your peers that you are able to intelligently propose and carry out an independent research plan that is sufficiently novel. (Cue grumbling from older PhDs about the dumbing down of modern degree programs and lack of intellectual rigor these days.)
So that's where the line is usually drawn. The PhDs design the experiments, and the Bachelors and Masters (and lower-ranking PhDs) carry them out. The Bachelors and Masters may do some routine post-analysis, but ultimately they hand the results back to the PhDs to interpret and plan the next round of experiments. It's not that you can't find positions where Bachelors or Masters are significantly involved in planning and analysis, it's just that if you want your primary responsibility to be planning, you should probably shoot for a PhD.