Stiffen up slanted deck railing
I'd try something simple first.  The problem is there's too much play in the fastening of the post boards to the joists below.  

Gluing, suggested in another answer, could work - it would take out the play - you'd just have to be careful to use a good strong outdoor rated glue. I am not sure the glue would hold up with the weight of the post and the length of the lever.  Even if it does hold up, gluing would make it difficult to maintain - you would have to cut the posts off the joists if any post needs to be replaced.  

The posts are attached to the joists with two carriage bolts.  Carriage bolts are inserted through drilled holes, and the top of the bolt has a square chunk under the cap that keeps it from spinning when you tighten it.  You only put a wrench on the nut.  You can't apply too much force without spinning the bolt.  

I'd replace those carriage bolts with hex head bolts, and use a fender washer under the bolt head and the nut.  With a hex bolt, you'll have a wrench on the bolt head and another on the nut and you can apply a lot of force.  The fender washer will spread the force over more area than a regular washer.  

I'd also loosen all the bolts, then replace them one by one.  Let the rail sag before you tighten everything so it's not sprung in place, it's already at rest when you tighten everything.