"How do I get my downline members to participate in my team contests?"
I always try to understand my self fist and than i try to understand the others.
In this particular case i would ask myself " why would I participate in such a contest".
Just ask yourself that simple question "Why would i take part in this contest of mine".
I believe that you will find an answer to that easily.
Once you do, share that answer trough email with your affiliates. And simply explain the rewards of participation and how it will be of benefit to them as participants.